Evaluate The Expression: ${ \left(\frac{7 5}{7 2}\right)^2 }$

$$

Introduction

When dealing with exponents and fractions, it's essential to understand the rules of exponentiation and how to simplify expressions. In this article, we will evaluate the expression $\left(\frac{7^5}{7^2}\right)^2$ using the properties of exponents and fractions.

Understanding Exponents and Fractions

Exponents are a shorthand way of writing repeated multiplication. For example, $7^3$ means $7 \times 7 \times 7$. Fractions, on the other hand, are a way of representing a part of a whole. In this case, we have a fraction with exponents in the numerator and denominator.

Simplifying the Expression

To simplify the expression, we need to apply the rules of exponentiation. The first rule we will use is the quotient of powers rule, which states that $\frac{a^m}{a^n} = a^{m-n}$.

Using this rule, we can simplify the expression as follows:

$$\left(\frac{7^5}{7^2}\right)^2 = \left(7^{5-2}\right)^2 = \left(7^3\right)^2$$

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it by applying the power of a power rule, which states that $(a^m)^n = a^{mn}$.

Using this rule, we can evaluate the expression as follows:

$$\left(7^3\right)^2 = 7^{3 \times 2} = 7^6$$

Conclusion

In conclusion, we have evaluated the expression $\left(\frac{7^5}{7^2}\right)^2$ using the properties of exponents and fractions. We simplified the expression by applying the quotient of powers rule and then evaluated it by applying the power of a power rule. The final result is $7^6$.

Additional Examples

To further illustrate the concept, let's consider a few more examples.

Example 1

Evaluate the expression $\left(\frac{2^4}{2^2}\right)^3$.

Using the quotient of powers rule, we can simplify the expression as follows:

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

Using the power of a power rule, we can evaluate the expression as follows:

$$\left(2^2\right)^3 = 2^{2 \times 3} = 2^6$$

Example 2

Evaluate the expression $\left(\frac{3^5}{3^3}\right)^2$.

Using the quotient of powers rule, we can simplify the expression as follows:

$$\left(\frac{3^5}{3^3}\right)^2 = \left(3^{5-3}\right)^2 = \left(3^2\right)^2$$

Using the power of a power rule, we can evaluate the expression as follows:

$$\left(3^2\right)^2 = 3^{2 \times 2} = 3^4$$

Final Thoughts

In conclusion, evaluating expressions with exponents and fractions requires a solid understanding of the rules of exponentiation and how to simplify expressions. By applying the quotient of powers rule and the power of a power rule, we can simplify and evaluate expressions with ease.

Common Mistakes to Avoid

When working with exponents and fractions, it's essential to avoid common mistakes. Here are a few to watch out for:

• Not simplifying the expression: Failing to simplify the expression can lead to incorrect results.
• Not applying the correct rule: Applying the wrong rule can lead to incorrect results.
• Not evaluating the expression: Failing to evaluate the expression can lead to incorrect results.

Tips for Success

To succeed in evaluating expressions with exponents and fractions, follow these tips:

• Understand the rules of exponentiation: Make sure you understand the rules of exponentiation, including the quotient of powers rule and the power of a power rule.
• Simplify the expression: Simplify the expression by applying the quotient of powers rule.
• Evaluate the expression: Evaluate the expression by applying the power of a power rule.
• Check your work: Double-check your work to ensure that you have applied the correct rules and obtained the correct result.

Conclusion

In conclusion, evaluating expressions with exponents and fractions requires a solid understanding of the rules of exponentiation and how to simplify expressions. By applying the quotient of powers rule and the power of a power rule, we can simplify and evaluate expressions with ease. Remember to avoid common mistakes and follow the tips for success to ensure that you obtain the correct result.

Introduction

Evaluating expressions with exponents and fractions can be a challenging task, but with the right understanding and techniques, it can be made easier. In this article, we will answer some common questions related to evaluating expressions with exponents and fractions.

Q1: What is the quotient of powers rule?

A1: The quotient of powers rule states that $\frac{a^m}{a^n} = a^{m-n}$. This rule allows us to simplify expressions by subtracting the exponents in the numerator and denominator.

Q2: How do I apply the quotient of powers rule?

A2: To apply the quotient of powers rule, simply subtract the exponents in the numerator and denominator. For example, $\frac{2^4}{2^2} = 2^{4-2} = 2^2$.

Q3: What is the power of a power rule?

A3: The power of a power rule states that $(a^m)^n = a^{mn}$. This rule allows us to simplify expressions by multiplying the exponents.

Q4: How do I apply the power of a power rule?

A4: To apply the power of a power rule, simply multiply the exponents. For example, $(2^2)^3 = 2^{2 \times 3} = 2^6$.

Q5: Can I simplify an expression with a negative exponent?

A5: Yes, you can simplify an expression with a negative exponent. To do this, simply rewrite the expression with a positive exponent. For example, $\frac{1}{2^3} = 2^{-3}$.

Q6: How do I evaluate an expression with a variable in the exponent?

A6: To evaluate an expression with a variable in the exponent, simply substitute the value of the variable into the expression. For example, if $x = 2$, then $2^{x+1} = 2^{2+1} = 2^3$.

Q7: Can I simplify an expression with a fraction in the exponent?

A7: Yes, you can simplify an expression with a fraction in the exponent. To do this, simply rewrite the expression with a single exponent. For example, $2^{\frac{1}{2}} = \sqrt{2}$.

Q8: How do I evaluate an expression with multiple exponents?

A8: To evaluate an expression with multiple exponents, simply apply the rules of exponentiation in the correct order. For example, $2^{3+2} = 2^5$.

Q9: Can I simplify an expression with a zero exponent?

A9: Yes, you can simplify an expression with a zero exponent. To do this, simply rewrite the expression as 1. For example, $2^0 = 1$.

Q10: How do I evaluate an expression with a negative exponent and a variable in the exponent?

A10: To evaluate an expression with a negative exponent and a variable in the exponent, simply substitute the value of the variable into the expression and then apply the rules of exponentiation. For example, if $x = 2$, then $2^{-x} = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}$.

Conclusion

In conclusion, evaluating expressions with exponents and fractions requires a solid understanding of the rules of exponentiation and how to simplify expressions. By applying the quotient of powers rule and the power of a power rule, we can simplify and evaluate expressions with ease. Remember to avoid common mistakes and follow the tips for success to ensure that you obtain the correct result.

Additional Resources

For further practice and review, we recommend the following resources:

• Math textbooks: Consult a math textbook for additional practice problems and examples.
• Online resources: Visit online resources such as Khan Academy, Mathway, and Wolfram Alpha for additional practice problems and examples.
• Practice problems: Complete practice problems to reinforce your understanding of evaluating expressions with exponents and fractions.

Final Thoughts

Evaluating expressions with exponents and fractions is a critical skill that requires practice and patience. By following the rules of exponentiation and simplifying expressions, you can become proficient in evaluating expressions with ease. Remember to avoid common mistakes and follow the tips for success to ensure that you obtain the correct result.