Evaluate The Expression:$\left(125^{\frac{4}{9}}\right)\left(125^{\frac{2}{9}}\right) =$

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Introduction

In mathematics, the concept of exponents is crucial in solving various types of problems. Exponents are used to represent the power or the index of a number. When we have an expression with exponents, we can simplify it using the rules of exponents. In this article, we will evaluate the expression $\left(125^{\frac{4}{9}}\right)\left(125^{\frac{2}{9}}\right)$ using the rules of exponents.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $a^m$ means $a$ multiplied by itself $m$ times. In the expression $\left(125^{\frac{4}{9}}\right)\left(125^{\frac{2}{9}}\right)$, we have two exponents with the same base, which is $125$. The first exponent is $\frac{4}{9}$ and the second exponent is $\frac{2}{9}$.

Applying the Product of Powers Rule

The product of powers rule states that when we multiply two numbers with the same base, we add their exponents. In this case, we can apply the product of powers rule to simplify the expression. The product of powers rule can be expressed as:

$a^m \cdot a^n = a^{m+n}$

Using this rule, we can simplify the expression $\left(125^{\frac{4}{9}}\right)\left(125^{\frac{2}{9}}\right)$ as follows:

$\left(125^{\frac{4}{9}}\right)\left(125^{\frac{2}{9}}\right) = 125^{\frac{4}{9} + \frac{2}{9}}$

Simplifying the Exponent

Now, we need to simplify the exponent $\frac{4}{9} + \frac{2}{9}$. To do this, we can add the fractions by finding a common denominator, which is $9$ in this case. The sum of the fractions is:

$\frac{4}{9} + \frac{2}{9} = \frac{6}{9}$

Simplifying the Fraction

The fraction $\frac{6}{9}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is $3$. The simplified fraction is:

$\frac{6}{9} = \frac{2}{3}$

Evaluating the Expression

Now that we have simplified the exponent, we can evaluate the expression $\left(125^{\frac{4}{9}}\right)\left(125^{\frac{2}{9}}\right)$. We can rewrite the expression as:

$125^{\frac{2}{3}}$

Understanding the Result

The expression $125^{\frac{2}{3}}$ can be evaluated by finding the cube root of $125$ and raising it to the power of $2$. The cube root of $125$ is $5$, so we can rewrite the expression as:

$5^2$

Evaluating the Final Expression

The final expression $5^2$ can be evaluated by multiplying $5$ by itself. The result is:

$25$

Conclusion

In this article, we evaluated the expression $\left(125^{\frac{4}{9}}\right)\left(125^{\frac{2}{9}}\right)$ using the rules of exponents. We applied the product of powers rule to simplify the expression and then simplified the exponent. Finally, we evaluated the expression to find the result, which is $25$.

Frequently Asked Questions

• What is the product of powers rule? The product of powers rule states that when we multiply two numbers with the same base, we add their exponents.
• How do we simplify the exponent $\frac{4}{9} + \frac{2}{9}$? We can add the fractions by finding a common denominator, which is $9$ in this case.
• What is the simplified fraction of $\frac{6}{9}$? The simplified fraction is $\frac{2}{3}$.
• How do we evaluate the expression $125^{\frac{2}{3}}$? We can find the cube root of $125$ and raise it to the power of $2$.

Final Answer

The final answer is $\boxed{25}$.

Introduction

In our previous article, we evaluated the expression $\left(125^{\frac{4}{9}}\right)\left(125^{\frac{2}{9}}\right)$ using the rules of exponents. In this article, we will answer some frequently asked questions related to evaluating expressions with exponents.

Q&A

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply two numbers with the same base, we add their exponents. This rule can be expressed as:

$a^m \cdot a^n = a^{m+n}$

Q: How do we simplify the exponent $\frac{4}{9} + \frac{2}{9}$?

A: We can add the fractions by finding a common denominator, which is $9$ in this case. The sum of the fractions is:

$\frac{4}{9} + \frac{2}{9} = \frac{6}{9}$

We can then simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $3$. The simplified fraction is:

$\frac{6}{9} = \frac{2}{3}$

Q: What is the simplified fraction of $\frac{6}{9}$?

A: The simplified fraction is $\frac{2}{3}$.

Q: How do we evaluate the expression $125^{\frac{2}{3}}$?

A: We can find the cube root of $125$ and raise it to the power of $2$. The cube root of $125$ is $5$, so we can rewrite the expression as:

$5^2$

We can then evaluate the final expression by multiplying $5$ by itself. The result is:

$25$

Q: What is the final answer to the expression $\left(125^{\frac{4}{9}}\right)\left(125^{\frac{2}{9}}\right)$?

A: The final answer is $\boxed{25}$.

Q: Can we use the product of powers rule to simplify expressions with different bases?

A: No, the product of powers rule only applies to expressions with the same base. If we have expressions with different bases, we cannot use the product of powers rule to simplify them.

Q: How do we simplify expressions with negative exponents?

A: We can simplify expressions with negative exponents by rewriting them with positive exponents. For example, $a^{-m}$ can be rewritten as $\frac{1}{a^m}$.

Q: Can we use the product of powers rule to simplify expressions with fractional exponents?

A: Yes, we can use the product of powers rule to simplify expressions with fractional exponents. For example, $a^{\frac{m}{n}} \cdot a^{\frac{p}{q}}$ can be simplified as:

$a^{\frac{m}{n} + \frac{p}{q}}$

Conclusion

In this article, we answered some frequently asked questions related to evaluating expressions with exponents. We covered topics such as the product of powers rule, simplifying exponents, and evaluating expressions with negative and fractional exponents. We hope that this article has been helpful in clarifying any doubts you may have had about evaluating expressions with exponents.

Final Answer

The final answer is $\boxed{25}$.