Which Expression Is Equivalent To $\left(\frac{125^2}{125^{\frac{4}{3}}}\right$\]?A. $\frac{1}{25}$ B. $\frac{1}{10}$ C. 10 D. 25

Introduction

In mathematics, simplifying expressions with exponents and fractions is a crucial skill that can be applied to a wide range of problems. In this article, we will focus on simplifying the expression $\left(\frac{125^2}{125^{\frac{4}{3}}}\right)$ and determine which of the given options is equivalent to it.

Understanding Exponents

Before we dive into simplifying the expression, let's review the basics of exponents. An exponent is a small number that is raised to the power of a larger number. For example, in the expression $a^b$, $a$ is the base and $b$ is the exponent. When we multiply two numbers with the same base, we add their exponents. For example, $a^b \cdot a^c = a^{b+c}$.

Simplifying the Expression

Now that we have a basic understanding of exponents, let's simplify the expression $\left(\frac{125^2}{125^{\frac{4}{3}}}\right)$. To simplify this expression, we can use the rule that states $\frac{a^b}{a^c} = a^{b-c}$.

Using this rule, we can rewrite the expression as:

$$\left(\frac{125^2}{125^{\frac{4}{3}}}\right) = 125^{2-\frac{4}{3}}$$

Evaluating the Exponent

Now that we have simplified the expression, let's evaluate the exponent. To do this, we need to subtract $\frac{4}{3}$ from $2$. To subtract fractions, we need to have a common denominator. In this case, the common denominator is $3$.

$$2-\frac{4}{3} = \frac{6}{3} - \frac{4}{3} = \frac{2}{3}$$

So, the expression can be rewritten as:

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

Simplifying the Expression Further

Now that we have evaluated the exponent, let's simplify the expression further. To do this, we can use the rule that states $a^{\frac{1}{n}} = \sqrt[n]{a}$. In this case, we can rewrite the expression as:

$$125^{\frac{2}{3}} = \sqrt[3]{125^2}$$

Evaluating the Expression

Now that we have simplified the expression, let's evaluate it. To do this, we need to find the cube root of $125^2$. To find the cube root of a number, we can raise it to the power of $\frac{1}{3}$.

$$\sqrt[3]{125^2} = (125^2)^{\frac{1}{3}} = 125^{\frac{2}{3}}$$

Which Option is Equivalent?

Now that we have simplified the expression, let's determine which of the given options is equivalent to it. The options are:

A. $\frac{1}{25}$ B. $\frac{1}{10}$ C. 10 D. 25

To determine which option is equivalent, we can evaluate each option and compare it to the simplified expression.

Option A: $\frac{1}{25}$

To evaluate option A, we can rewrite it as a fraction with a denominator of $25$.

$$\frac{1}{25} = \frac{1}{5^2}$$

Since the denominator is $5^2$, we can rewrite the expression as:

$$\frac{1}{5^2} = \frac{1}{5^2} \cdot \frac{5^{-2}}{5^{-2}} = 5^{-4}$$

Now that we have rewritten the expression, let's compare it to the simplified expression.

$$5^{-4} \neq 125^{\frac{2}{3}}$$

So, option A is not equivalent to the simplified expression.

Option B: $\frac{1}{10}$

To evaluate option B, we can rewrite it as a fraction with a denominator of $10$.

$$\frac{1}{10} = \frac{1}{2 \cdot 5}$$

Since the denominator is $2 \cdot 5$, we can rewrite the expression as:

$$\frac{1}{2 \cdot 5} = \frac{1}{2} \cdot \frac{1}{5}$$

Now that we have rewritten the expression, let's compare it to the simplified expression.

$$\frac{1}{2} \cdot \frac{1}{5} \neq 125^{\frac{2}{3}}$$

So, option B is not equivalent to the simplified expression.

Option C: 10

To evaluate option C, we can rewrite it as a fraction with a denominator of $1$.

$$10 = \frac{10}{1}$$

Since the denominator is $1$, we can rewrite the expression as:

$$\frac{10}{1} = 10^1$$

Now that we have rewritten the expression, let's compare it to the simplified expression.

$$10^1 \neq 125^{\frac{2}{3}}$$

So, option C is not equivalent to the simplified expression.

Option D: 25

To evaluate option D, we can rewrite it as a fraction with a denominator of $1$.

$$25 = \frac{25}{1}$$

Since the denominator is $1$, we can rewrite the expression as:

$$\frac{25}{1} = 25^1$$

Now that we have rewritten the expression, let's compare it to the simplified expression.

$$25^1 \neq 125^{\frac{2}{3}}$$

So, option D is not equivalent to the simplified expression.

Conclusion

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Q: What is the rule for simplifying expressions with exponents?

A: The rule for simplifying expressions with exponents is $\frac{a^b}{a^c} = a^{b-c}$. This rule allows us to simplify expressions by subtracting the exponents.

Q: How do I evaluate an exponent?

A: To evaluate an exponent, you need to subtract the exponents. For example, if you have the expression $a^{b-c}$, you need to subtract $c$ from $b$ to get the new exponent.

Q: What is the difference between a base and an exponent?

A: A base is the number that is being raised to a power, and an exponent is the power to which the base is being raised. For example, in the expression $a^b$, $a$ is the base and $b$ is the exponent.

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

A: To simplify an expression with a cube root, you can use the rule that states $a^{\frac{1}{n}} = \sqrt[n]{a}$. This rule allows you to rewrite the cube root as a fractional exponent.

Q: What is the relationship between exponents and fractions?

A: Exponents and fractions are related in that you can rewrite a fraction as a decimal or a percentage by raising the base to a power. For example, the fraction $\frac{1}{2}$ can be rewritten as $0.5$ or $50\%$.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you need to use the rule that states $a^b \cdot a^c = a^{b+c}$. This rule allows you to combine the exponents by adding them.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent is an exponent that is greater than zero, and a negative exponent is an exponent that is less than zero. For example, in the expression $a^b$, if $b$ is positive, then the exponent is positive. If $b$ is negative, then the exponent is negative.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can use the rule that states $a^{-b} = \frac{1}{a^b}$. This rule allows you to rewrite the negative exponent as a fraction.

Q: What is the relationship between exponents and logarithms?

A: Exponents and logarithms are related in that they are inverse operations. For example, if you have the expression $a^b$, you can take the logarithm of both sides to get $b = \log_a(a^b)$.

Q: How do I simplify an expression with a logarithm?

A: To simplify an expression with a logarithm, you can use the rule that states $\log_a(a^b) = b$. This rule allows you to rewrite the logarithm as an exponent.

Conclusion

In conclusion, simplifying expressions with exponents and fractions is a crucial skill that can be applied to a wide range of problems. By understanding the rules for simplifying expressions, you can simplify complex expressions and solve problems more efficiently.