How We Teach Exponents and Powers to Kids

About the series: Python for kids

In our endeavour to make learning fun and free of memorization, we have been using Python to allow children to experiment, conceptualize, form their own opinion and discover facts. This way, children engage with the learning process with more intensity and the they show higher retention rate vis a vis conventional learning. Our results have shown that the children learning with us, ranging from grade 3 to graduate students, have a recall value better than conventional learning, making the learning many times faster.

Our schools focus on rewarding the students who score better and in turn reward the teachers and schools whose students score better. This leads to a system that makes sure that students score higher and higher marks, leading to rote learning and conceptually weaker students.

We have created program and content to stimulate and challenge their problem-solving abilities. This makes them think and try different things before they get it right. Our focus is on making them work harder to get things right, not to worry about failing many times before they succeed and trying to a few solve complex problems rather than practising too many easy and pattern-based problems.

The focus shifts from being always ‘right’ to getting it right after several failed attempts.

Programming Interface vs App or Website

The apps and websites are restrictive in nature and do not allow students to experiment in same way as the direct programming interfaces like Jupyter notebooks provide. That is why we have launched our services with direct programming interface.

Self-learning vs Instructor-assisted

Though these programs are created to help students do self-learning, it is best suited to be used as instructor-assisted learning. We believe that it is to naive to think self-learning can be stimulating enough to motivate students try challenging problems and keep motivation alive even after several failed attemts on a regular basis.

Our software can be used for self-learning but the full potential is realized when used with an instructor. Ask the author for more details or put comments.

Exponent or Power

Exponentiation is a mathematical operation, written as bⁿ, involving two numbers, the base b and the exponent or power n, and pronounced as "b raised to the power of n". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bⁿ is the product of multiplying n bases.

Isn't the above definition complex and difficult to understand? And of course it is. So we we also promote intutive learning.

• If we have to calculate 5 times 5, we write 5 * 5.
• If we have to calculate 5 times 5 times 5, we write 5 5 5.
• If we have to calculate 5 times 5 times 5 ... and repeat this process 100 times, we write 5 5 5 * ... 100 times.
• But there is a better way to do this: 5¹⁰⁰

Our Packages

We use our own python packages, but we emphasize that children write code without using our packages as well. The objective of our packages is to focus on conceptual learning rather than programming. It also allows us to bridge that gap between something is required and not available out of box.

Let us get started

We have created a package for exponents or power. Let us import it.

from xv.math.basicmaths import PowerManager
from IPython.display import HTML

ke = PowerManager()

ke.getRandomProblem(problem_type = 3)

Simplify as power of 8
$8 * 8$

ke.printAnswer()

$8^{2}$

ke.printSolution()

$8 * 8$

= $8^{2}$

The power is simply count of number of times a number is multipled.

It is read as 8 raised to the power 2.
In short, 8 power 2.

8 is called base.
2 has several names like power, exponent.

As you see, the questions are randomized and are created by our AI/ML models which analyze difficulty levels, the skill status of students and learning objectives to make it the learning as relevant, fast and conceptual as possible.

The problems are not repeated, allowing students or group of students to repeat problems without getting used to patterns. During the tests, all students get differnt questions that relate to same skill levels, making it fair and conceptual test, rather than a test of practice and memorization.

If we run the same problem type again, we get the following question:

ke.getRandomProblem(problem_type = 3)

Simplify as power of 4
$4 * 4 * 4 * 4 * 4 * 4$

ke.printAnswer()

$4^{6}$

ke.printSolution()

$4 * 4 * 4 * 4 * 4 * 4$

= $4^{6}$

The power is simply count of number of times a number is multipled.

It is read as 4 raised to the power 6.
In short, 4 power 6.

4 is called base.
6 has several names like power, exponent.

A gradual conceptual climbing

Like a game with several levels

Let us run for more problem types. You will note that all problems are provided with full solutions. We keep the details to adequate levels so that students discover the concept, rather than read and learn.

ke = PowerManager()
for i in range(len(ke._problemTemplates)):
    display(HTML(f"<h2>Question {i}</h2>"))
    ke.getRandomProblem(problem_type = i)
    display(ke.printProblem())

    display(HTML(f"<h6>Answer:</h6>"))
    display(ke.printAnswer())

    display(HTML(f"<h6>Solution:</h6>"))
    display(ke.printSolution())
    print(f"""{'-'*50}
    """)
    pass

Question 0

Use exponent or power to do the following:

Start at $8$

1.      From $8$, go to $\frac{1}{8}$

2.      From $\displaystyle \left(\frac{1}{8}\right)$, go to $\displaystyle \frac{1}{2097152}$

3.      From $\left(\frac{1}{2097152}\right)$, go to $\frac{1}{8}$

4.      From $\left(\frac{1}{8}\right)$, go to $8$

5.      From $\displaystyle 8$, go to $\displaystyle 2097152$

6.      From $\left(2097152\right)$, go to $\frac{1}{2097152}$

7.      From $\left(\frac{1}{2097152}\right)$, go to $2097152$

8.      From $\left(2097152\right)$, go to $8$

9.      From $8$, go to $\frac{1}{2097152}$

10.      From $\left(\frac{1}{2097152}\right)$, go to $8$

Answer:

1.      ${8}^{-1}$

2.      ${\left(\frac{1}{8}\right)}^{7}$

3.      ${\left(\frac{1}{2097152}\right)}^{\frac{1}{7}}$

4.      ${\left(\frac{1}{8}\right)}^{-1}$

5.      ${8}^{7}$

6.      ${\left(2097152\right)}^{-1}$

7.      ${\left(\frac{1}{2097152}\right)}^{-1}$

8.      ${\left(2097152\right)}^{\frac{1}{7}}$

9.      ${8}^{-7}$

10.      ${\left(\frac{1}{2097152}\right)}^{- \frac{1}{7}}$

Solution:

Start at $8$


1.      From $8$, go to $\frac{1}{8}$


         ${8}^{-1} = \frac{1}{8}$


2.      From $\displaystyle \left(\frac{1}{8}\right)$, go to $\displaystyle \frac{1}{2097152}$


         $\displaystyle
{\left(\frac{1}{8}\right)}^{7}
= \left(\frac{1}{8}\right) * \left(\frac{1}{8}\right) * \left(\frac{1}{8}\right) * \left(\frac{1}{8}\right) * \left(\frac{1}{8}\right) * \left(\frac{1}{8}\right) * \left(\frac{1}{8}\right) = \frac{1}{2097152}
$


3.      From $\left(\frac{1}{2097152}\right)$, go to $\frac{1}{8}$


         ${\left(\frac{1}{2097152}\right)}^{\frac{1}{7}}
= {\left(\frac{1}{8}\right)}^{7 * \frac{1}{7}}
= \frac{1}{8}$


4.      From $\left(\frac{1}{8}\right)$, go to $8$


         ${\left(\frac{1}{8}\right)}^{-1} = 8$


5.      From $\displaystyle 8$, go to $\displaystyle 2097152$


         $\displaystyle
{8}^{7}
= 8 * 8 * 8 * 8 * 8 * 8 * 8 = 2097152
$


6.      From $\left(2097152\right)$, go to $\frac{1}{2097152}$


         ${\left(2097152\right)}^{-1} = \frac{1}{2097152}$


7.      From $\left(\frac{1}{2097152}\right)$, go to $2097152$


         ${\left(\frac{1}{2097152}\right)}^{-1} = 2097152$


8.      From $\left(2097152\right)$, go to $8$


         ${\left(2097152\right)}^{\frac{1}{7}}
= {\left(8\right)}^{7 * \frac{1}{7}}
= 8$


9.      From $8$, go to $\frac{1}{2097152}$


         ${8}^{-7}
= { \frac { 1 } { {8}^{7} } }
= \frac{1}{2097152}$


10.      From $\left(\frac{1}{2097152}\right)$, go to $8$


         ${\left(\frac{1}{2097152}\right)}^{- \frac{1}{7}}
= {\left(8\right)}^{7 * - \frac{1}{7}}
= 8$

--------------------------------------------------
    

Question 1

Find value of
$4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4$

Answer:

$262144$

Solution:

$4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4$

= $262144$

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Question 2

$9 * 9 * 9 * 9 * 9 * 9$ is written as $9^{6}$. What would you write for
$7 * 7 * 7 * 7$

Answer:

$7^{4}$

Solution:

$7 * 7 * 7 * 7$

= $7^{4}$

The power is simply count of number of times a number is multipled.

It is read as 7 raised to the power 4.
In short, 7 power 4.

7 is called base.
4 has several names like power, exponent.

--------------------------------------------------
    

Question 3

Simplify as power of 4
$4 * 4 * 4$

Answer:

$4^{3}$

Solution:

$4 * 4 * 4$

= $4^{3}$

The power is simply count of number of times a number is multipled.

It is read as 4 raised to the power 3.
In short, 4 power 3.

4 is called base.
3 has several names like power, exponent.

--------------------------------------------------
    

Question 4

Simplify as power of 6
(6 * 6 * 6) * (6 * 6 * 6 * 6)

Answer:

$6^{7}$

Solution:

(6 * 6 * 6) * (6 * 6 * 6 * 6)

= $6^{3} * 6^{4}$

= $6^{3 + 4}$

= $6^{7}$

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Question 5

Find value of
3 * 3 * 3 * .... 81 times

Answer:

$3^{81}$

Solution:

3 * 3 * 3 * .... 81 times

= $3^{81}$

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Question 6

Simplify as power of 5

(5 * 5 * 5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5 * 5 * 5)

Answer:

$5^{28}$

Solution:

(5 * 5 * 5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5 * 5 * 5) * (5 * 5 * 5 * 5 * 5 * 5 * 5)

= $5^7$ * $5^7$ * $5^7$ * $5^7$

= $5^{7 * 4}$

= $5^{28}$

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Question 7

Simplify as power of 7

$\displaystyle { \frac { 7 * 7 * 7 * 7 * 7 } { 7 * 7 * 7 * 7 * 7 * 7 } } $

Answer:

$\displaystyle 7^{-1}$

Solution:

$\displaystyle { \frac { 7 * 7 * 7 * 7 * 7 } { 7 * 7 * 7 * 7 * 7 * 7 } } $

$\displaystyle = { \frac { 7^{5} } { 7^{6} } } $

$\displaystyle = 7^{5 - 6}$          As the numerator powers are added, denominator powers will be subtracted.

$\displaystyle = 7^{-1}$

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Question 8

What is value of

$\displaystyle \left(\frac{4}{3}\right)^{-\infty}$

Answer:

$\displaystyle \text{Undefined}$

Solution:

$\displaystyle \infty$ is undefined. Hence,

$\displaystyle \implies \left(\frac{4}{3}\right)^{-\infty} = \text{Undefined}$

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Question 9

What is value of

$\displaystyle \left(-\infty\right)^{0}$

Answer:

$\displaystyle \text{Undefined}$

Solution:

$\displaystyle
{ \frac { -\infty } { -\infty } }
= { \frac { \left(-\infty\right)^{1} } { \left(-\infty\right)^{1} } }
= {\left(-\infty\right)}^{1-1}
= \left(-\infty\right)^{0}
$

$\displaystyle \implies \text{Undefined} = \left(-\infty\right)^{0}$

$\displaystyle \implies \left(-\infty\right)^{0} = \text{Undefined}$

--------------------------------------------------
    

Question 10

What is value of

$\displaystyle \left(-1\right)^{-5}$

Answer:

$\displaystyle { \frac { 1 } { \left(-1\right)^{5} } } $

Solution:

$\displaystyle
{ \frac { 1 } { \left(-1\right)^{5} } }
= { \frac { {\left(-1\right)}^{0} } { \left(-1\right)^{5} } }
= {\left(-1\right)}^{0 - -5}
= \left(-1\right)^{-5}
$

$\displaystyle \implies { \frac { 1 } { \left(-1\right)^{5} } } = \left(-1\right)^{-5}$

$\displaystyle \implies \left(-1\right)^{-5} = { \frac { 1 } { \left(-1\right)^{5} } } $

--------------------------------------------------
    

Question 11

Express $1^{-2}$ as as rational number.

Answer:

$ { \frac { 1 } { 1 } } $

Solution:

$1^{-2}$

The power becomes negative when we move it from numerator to denominator or from denominator to numerator
$= { \frac { 1 } { 1^{2} } } $

$= { \frac { 1 } { 1 } } $

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Question 12

Express $ { \frac { 1 } { 7^{-3} } } $ as power of $7$.

Answer:

$7^{3}$

Solution:

$ { \frac { 1 } { 7^{-3} } } $

The power becomes negative when we move it from numerator to denominator or from denominator to numerator

$= 7^{--3}$

$= 7^{3}$

Alternatively:
$ { \frac { 1 } { 7^{-3} } } $

$= { \frac { 7^0 } { 7^{-3} } } $

$= 7^{0 - -3} $

$= 7^{3}$

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Question 13

Simplify as power of 7
$\displaystyle 7 * 7 * 7 * 7 * 7 * 7 * 7 * { \frac { 7 * 7 * 7 * 7 } { 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 } } $

Answer:

$\displaystyle 7^{2}$

Solution:

$\displaystyle 7 * 7 * 7 * 7 * 7 * 7 * 7 * { \frac { 7 * 7 * 7 * 7 } { 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 } } $

$\displaystyle = { \frac { 7^{7} * 7^{4} } { 7^{9} } } $

$\displaystyle = 7^{7 + 4 - 9}$

$\displaystyle = 7^{2}$

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Question 14

Simplify as power of 4

$\displaystyle { \frac { 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 } { 4 * 4 * 4 * 4 } } * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4$

Answer:

$\displaystyle 4^{13}$

Solution:

$\displaystyle { \frac { 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 } { 4 * 4 * 4 * 4 } } * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4$

$\displaystyle = { \frac { 4^{8} } { 4^{4} } } * 4^{9}$

$\displaystyle = 4^{8 + 9 - 4}$

$\displaystyle = 4^{13}$

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Question 15

Simplify as power of 4

$(4 * 4 * 4 * 4 * 4 * 4)^{5}$

Answer:

$4^{30}$

Solution:

$(4 * 4 * 4 * 4 * 4 * 4)^{5}$

= $\left(4^{6}\right)^{5}$

= $4^{6 * 5}$

= $4^{30}$

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Question 16

Simplify
$\displaystyle 1$

Answer:

$\displaystyle 1$

Solution:

$\displaystyle 1$

$\displaystyle = 1$

$\displaystyle = \left( {1} \right)^{2 * - \frac{1}{2}}$

$\displaystyle = \left( {1} \right)^{-1}$

$\displaystyle = { \frac { 1 } { 1 } } $

$\displaystyle = 1$

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Question 17

Simplify as power of 5

$(5 * 5 * 5 * 5 * 5 * 5 * 5)^{6} * (5 * 5 * 5 * 5 * 5)^{2}$

Answer:

${5}^{52}$

Solution:

$(5 * 5 * 5 * 5 * 5 * 5 * 5)^{6} * (5 * 5 * 5 * 5 * 5)^{2}$

= $ { \left({5}^{7}\right) }^{6} * { \left({5}^{5}\right) }^{2} $

= ${5}^{7 * 6} * {5}^{5 * 2}$

= ${5}^{42} * {5}^{10}$

= ${5}^{42 + 10}$

= ${5}^{52}$

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Question 18

Simplify as power of 8

$\displaystyle { \frac { (8 * 8 * 8)^{4} } { (8 * 8 * 8 * 8 * 8 * 8 * 8 * 8 * 8)^{8} } } $

Answer:

$\displaystyle {8}^{-60}$

Solution:

$\displaystyle { \frac { (8 * 8 * 8)^{4} } { (8 * 8 * 8 * 8 * 8 * 8 * 8 * 8 * 8)^{8} } } $

$\displaystyle = { \frac { { \left({8}^{3}\right) }^{4} } { { \left({8}^{9}\right) }^{8} } } $

$\displaystyle = { \frac { {8}^{3 * 4} } { {8}^{9 * 8} } } $

$\displaystyle = { \frac { {8}^{12} } { {8}^{72} } } $

$\displaystyle = {8}^{12 - 72}$

$\displaystyle = {8}^{-60}$

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Question 19

Simplify as power of 8

$\displaystyle { \left(8^{5}\right) }^{4}$

Answer:

$\displaystyle 8^{20}$

Solution:

$\displaystyle { \left(8^{5}\right) }^{4}$

$\displaystyle = 8^{5 * 4}$

$\displaystyle = 8^{20}$

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Question 20

Simplify as power of 4

$\displaystyle { \left(4^{4}\right) }^{ { \frac { 1 } { 9 } } }$

Answer:

$\displaystyle 4^{ { \frac { 4 } { 9 } } }$

Solution:

$\displaystyle { \left(4^{4}\right) }^{ { \frac { 1 } { 9 } } }$

$\displaystyle = 4^{4 * { \frac { 1 } { 9 } } }$

$\displaystyle = 4^{ { \frac { 4 } { 9 } } }$

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Question 21

Simplify as power of 4

$\displaystyle \left({ \left(4 * 4 * 4 * 4\right)^{3} * \left(4 * 4 * 4 * 4 * 4\right)^{9} }\right) ^{2}$

Answer:

$\displaystyle = 4^{114}$

Solution:

$\displaystyle \left({ \left(4 * 4 * 4 * 4\right)^{3} * \left(4 * 4 * 4 * 4 * 4\right)^{9} }\right) ^{2}$

$\displaystyle = \left({ \left(4^{4}\right)^{3} * \left(4^{5}\right)^{9} }\right) ^{2}$

$\displaystyle = \left(4^{4 * 3} * 4^{5 * 9}\right) ^{2}$

$\displaystyle = \left(4^{12} * 4^{45}\right) ^{2}$

$\displaystyle = { \left(4^{12 + 45}\right) } ^ { 2 }$

$\displaystyle = { \left(4^{57}\right) } ^ { 2 }$

$\displaystyle = 4^{57 * 2}$

$\displaystyle = 4^{114}$

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Question 22

Simplify as power of 8

$\displaystyle \left({ { \frac { (8 * 8 * 8 * 8 * 8 * 8 * 8 * 8 * 8)^{4} } { (8 * 8 * 8)^{5} } } }\right) ^{8}$

Answer:

$\displaystyle 8^{168}$

Solution:

$\displaystyle \left({ { \frac { (8 * 8 * 8 * 8 * 8 * 8 * 8 * 8 * 8)^{4} } { (8 * 8 * 8)^{5} } } }\right) ^{8}$

$\displaystyle = \left({ { \frac { \left(8^{9}\right)^{4} } { \left(8^{3}\right)^{5} } } }\right) ^{8}$

$\displaystyle = \left({ { \frac { 8^{9 * 4} } { 8^{3 * 5} } } }\right) ^{8}$

$\displaystyle = \left({ { \frac { 8^{36} } { 8^{15} } } }\right) ^{8}$

$\displaystyle = { \left(8^{36 - 15}\right) } ^ { 8 }$

$\displaystyle = { \left(8^{21}\right) } ^ { 8 }$

$\displaystyle = 8^{21 * 8}$

$\displaystyle = 8^{168}$

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Question 23

Simplify as power of 5

$\displaystyle \left(5^{7} * 5^{4}\right)^{6}$

Answer:

$\displaystyle 5^{66}$

Solution:

$\displaystyle \left(5^{7} * 5^{4}\right)^{6}$

$\displaystyle = \left(5^{7 + 4}\right)^{6}$

$\displaystyle = \left(5^{11}\right)^{6}$

$\displaystyle = 5^{11 * 6}$

$\displaystyle = 5^{66}$

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Question 24

Simplify as power of 2

$\displaystyle \left( { \frac { 2^{8} } { 2^{9} } } \right)^{5}$

Answer:

$\displaystyle 2^{-5}$

Solution:

$\displaystyle \left( { \frac { 2^{8} } { 2^{9} } } \right)^{5}$

$\displaystyle = \left(2^{8 - 9}\right)^{5}$

$\displaystyle = \left(2^{-1}\right)^{5}$

$\displaystyle = 2^{-1 * 5}$

$\displaystyle = 2^{-5}$

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Question 25

Express as power of prime numbers:

$\displaystyle -4913^4$

Answer:

$\displaystyle {\left(-17\right)}^{ 12 } $

Solution:

$\displaystyle -4913^4$

$\displaystyle = {\left({\left(-17\right)}^3\right)^4} $

$\displaystyle = {\left({\left(-17\right)}^3\right)^4} $

$\displaystyle = {\left(-17\right)}^{ 3 * 4 } $

$\displaystyle = {\left(-17\right)}^{ 12 } $

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Question 26

Express as power of prime numbers

$\displaystyle 22^5$

Answer:

$\displaystyle 2^5 * 11^5 $

Solution:

$\displaystyle 22^5$

$\displaystyle = {\left(2*11\right)^5} $

$\displaystyle = 2^5 * 11^5 $

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Question 27

Express as power of prime numbers

$527182965101^3$

Answer:

$\displaystyle 13^{ 15 } * 17^{ 15 } $

Solution:

$\displaystyle 527182965101^3$

$\displaystyle = {\left(13^5 * 17^5\right)^3} $

$\displaystyle = {\left(13^5\right)^3} * {\left(17^5\right)^3} $

$\displaystyle = 13^{ 5 * 3 } * 17^{ 5 * 3 } $

$\displaystyle = 13^{ 15 } * 17^{ 15 } $

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Question 28

Find value of x

$x^{2} = 49 $

Answer:

$x = 7$

Solution:

$x^{2} = 49 $

$\displaystyle \implies x^{2} = 7^2 $

$\displaystyle \implies (x^{2})^{1/2} = (7^2)^{1/2} $

$\displaystyle \implies x = 7^{2/2}$

$\displaystyle \implies x = 7$

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Question 29

Find value of x

$7^{x} = 2401 $

Answer:

$\displaystyle 4$

Solution:

$7^{x} = 2401 $

$\displaystyle \implies 7^{x} = 7^4 $

$\displaystyle \implies x = 4$

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Our experience so far

We have been running several classes and sessions and the above problems are covered generally in two one our sessions with students who are learning the exponents for the first time.

We repeat the concept after a few weeks without any revision in-between. The retention rate is 80-90% which is pretty high.

How to sustain retention of concepts?

Out AI/ML models create network of concepts, we call them Knowledge Elements(KE). Networked KEs are used to use already learned concepts in problem solving of related concepts or higher level concepts. This way, they keep on using the concepts already learned without practicing or slowing down the learning process.