Prime Factorization of 300

Oct 13, 2025 by windailys.com 108 views

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Introduction

In this article, we will evaluate three given mathematical expressions under specific assumptions. These expressions involve square roots and cube roots, and we will simplify them step by step to obtain the final results.

Expression 1: $\sqrt{300}$ under the assumption $14 \sqrt{6}$

To evaluate the expression $\sqrt{300}$ , we can start by factoring the number 300 into its prime factors.

Prime Factorization of 300

The prime factorization of 300 is:

300 = 2 × 2 × 3 × 5 × 5

Simplifying the Square Root

Now, we can simplify the square root of 300 by grouping the prime factors into pairs.

$\sqrt{300}$ = $\sqrt{2 × 2 × 3 × 5 × 5}$

= $\sqrt{(2 × 2) × (3 × 5 × 5)}$

= $\sqrt{4 × 75}$

= $\sqrt{4} × \sqrt{75}$

= 2 × $\sqrt{75}$

Further Simplification

We can further simplify the square root of 75 by factoring it into its prime factors.

75 = 3 × 5 × 5

So, $\sqrt{75}$ = $\sqrt{3 × 5 × 5}$

= $\sqrt{3} × \sqrt{5 × 5}$

= $\sqrt{3} × 5$

Now, we can substitute this back into our original expression.

$\sqrt{300}$ = 2 × $\sqrt{75}$

= 2 × ( $\sqrt{3} × 5$ )

= 2 × $\sqrt{3}$ × 5

= 10 $\sqrt{3}$

Conclusion

Therefore, under the assumption $14 \sqrt{6}$ , the value of $\sqrt{300}$ is 10 $\sqrt{3}$ .

Expression 2: $\sqrt{3}$ under the assumption $4 \sqrt{2}$

To evaluate the expression $\sqrt{3}$ , we can start by simplifying the assumption $4 \sqrt{2}$ .

Simplifying the Assumption

We can simplify the assumption $4 \sqrt{2}$ by multiplying the number 4 by the square root of 2.

$4 \sqrt{2}$ = 4 × $\sqrt{2}$

= $\sqrt{2} × \sqrt{2} × 4$

= $\sqrt{2^2} × 4$

= 2 × 4

= 8

Conclusion

Therefore, under the assumption $4 \sqrt{2}$ , the value of $\sqrt{3}$ is simply $\sqrt{3}$ , as the assumption does not affect the value of the expression.

Expression 3: $\sqrt[3]{\sqrt{196}}$ under the assumption $\sqrt[3]{\sqrt{2}}$

To evaluate the expression $\sqrt[3]{\sqrt{196}}$ , we can start by simplifying the square root of 196.

Simplifying the Square Root

We can simplify the square root of 196 by factoring it into its prime factors.

196 = 2 × 2 × 7 × 7

So, $\sqrt{196}$ = $\sqrt{2 × 2 × 7 × 7}$

= $\sqrt{(2 × 2) × (7 × 7)}$

= $\sqrt{4 × 49}$

= $\sqrt{4} × \sqrt{49}$

= 2 × $\sqrt{49}$

Further Simplification

We can further simplify the square root of 49 by factoring it into its prime factors.

49 = 7 × 7

So, $\sqrt{49}$ = $\sqrt{7 × 7}$

= $\sqrt{7} × \sqrt{7}$

= 7

Now, we can substitute this back into our original expression.

$\sqrt{196}$ = 2 × $\sqrt{49}$

= 2 × 7

= 14

Conclusion

Therefore, under the assumption $\sqrt[3]{\sqrt{2}}$ , the value of $\sqrt[3]{\sqrt{196}}$ is $\sqrt[3]{14}$ .

Conclusion

In this article, we evaluated three given mathematical expressions under specific assumptions. We simplified each expression step by step to obtain the final results. The expressions involved square roots and cube roots, and we used prime factorization and simplification techniques to evaluate them.

Final Results

$\sqrt{300}$ under the assumption $14 \sqrt{6}$ = 10 $\sqrt{3}$
$\sqrt{3}$ under the assumption $4 \sqrt{2}$ = $\sqrt{3}$
$\sqrt[3]{\sqrt{196}}$ under the assumption $\sqrt[3]{\sqrt{2}}$ = $\sqrt[3]{14}$

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Introduction

In our previous article, we evaluated three given mathematical expressions under specific assumptions. We simplified each expression step by step to obtain the final results. In this article, we will answer some frequently asked questions (FAQs) related to evaluating mathematical expressions.

Q: What is the difference between a square root and a cube root?

A: A square root is a mathematical operation that finds the number that, when multiplied by itself, gives a specified value. For example, the square root of 16 is 4, because 4 × 4 = 16. A cube root, on the other hand, is a mathematical operation that finds the number that, when multiplied by itself twice, gives a specified value. For example, the cube root of 64 is 4, because 4 × 4 × 4 = 64.

Q: How do I simplify a square root expression?

A: To simplify a square root expression, you can start by factoring the number inside the square root into its prime factors. Then, you can group the prime factors into pairs and simplify the expression.

Q: What is the assumption in evaluating a mathematical expression?

A: The assumption in evaluating a mathematical expression is a given value or condition that is used to simplify the expression. For example, in the expression $\sqrt{300}$ under the assumption $14 \sqrt{6}$ , the assumption is $14 \sqrt{6}$ , which is used to simplify the expression.

Q: How do I evaluate an expression with a cube root?

A: To evaluate an expression with a cube root, you can start by simplifying the cube root expression. Then, you can substitute the simplified expression back into the original expression and simplify further.

Q: What is the difference between a rational and irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., a/b, where a and b are integers and b is not zero. An irrational number, on the other hand, is a number that cannot be expressed as the ratio of two integers. For example, the square root of 2 is an irrational number, because it cannot be expressed as a ratio of two integers.

Q: How do I determine if a number is rational or irrational?

A: To determine if a number is rational or irrational, you can try to express it as a ratio of two integers. If you can express it as a ratio of two integers, then it is a rational number. If you cannot express it as a ratio of two integers, then it is an irrational number.

Q: What is the importance of evaluating mathematical expressions?

A: Evaluating mathematical expressions is important because it helps us to simplify complex expressions and understand the underlying mathematical concepts. It also helps us to solve real-world problems and make informed decisions.

Q: How can I practice evaluating mathematical expressions?

A: You can practice evaluating mathematical expressions by working on problems and exercises that involve simplifying and evaluating expressions. You can also try to simplify and evaluate expressions on your own, using the techniques and strategies that we discussed in this article.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to evaluating mathematical expressions. We discussed the difference between a square root and a cube root, how to simplify a square root expression, the assumption in evaluating a mathematical expression, how to evaluate an expression with a cube root, the difference between a rational and irrational number, how to determine if a number is rational or irrational, and the importance of evaluating mathematical expressions. We also provided some tips and resources for practicing evaluating mathematical expressions.