Evaluate. Write Your Answers As Fractions.${ \begin{array}{l} \left(-\frac{4}{3}\right)^2 = \square \ -\left(\frac{5}{4}\right)^4 = \square \end{array} }$

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Understanding the Problem

When evaluating expressions with exponents, it's essential to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction). In this case, we have two expressions to evaluate: (−43)2\left(-\frac{4}{3}\right)^2 and −(54)4-\left(\frac{5}{4}\right)^4. We will break down each expression step by step to find the final answer.

Evaluating the First Expression

The first expression is (−43)2\left(-\frac{4}{3}\right)^2. To evaluate this, we need to follow the order of operations and first deal with the exponent. When a negative number is raised to an even power, the result is always positive.

Simplifying the Expression

(−43)2=(43)2\left(-\frac{4}{3}\right)^2 = \left(\frac{4}{3}\right)^2

Now, we can simplify the expression by squaring the fraction.

Squaring the Fraction

(43)2=4232\left(\frac{4}{3}\right)^2 = \frac{4^2}{3^2}

Using the exponent rule (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, we can simplify the expression further.

Simplifying the Fraction

4232=169\frac{4^2}{3^2} = \frac{16}{9}

So, the final answer to the first expression is 169\frac{16}{9}.

Evaluating the Second Expression

The second expression is −(54)4-\left(\frac{5}{4}\right)^4. To evaluate this, we need to follow the order of operations and first deal with the exponent. When a negative number is raised to an even power, the result is always positive. However, in this case, we have a negative sign in front of the expression, so the result will be negative.

Simplifying the Expression

−(54)4=−(5444)-\left(\frac{5}{4}\right)^4 = -\left(\frac{5^4}{4^4}\right)

Now, we can simplify the expression by raising the fraction to the power of 4.

Raising the Fraction to the Power of 4

−(5444)=−5444-\left(\frac{5^4}{4^4}\right) = -\frac{5^4}{4^4}

Using the exponent rule (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, we can simplify the expression further.

Simplifying the Fraction

−5444=−625256-\frac{5^4}{4^4} = -\frac{625}{256}

So, the final answer to the second expression is −625256-\frac{625}{256}.

Conclusion

In this article, we evaluated two expressions with exponents and wrote our answers as fractions. We followed the order of operations and simplified each expression step by step to find the final answer. The first expression was (−43)2\left(-\frac{4}{3}\right)^2, and the final answer was 169\frac{16}{9}. The second expression was −(54)4-\left(\frac{5}{4}\right)^4, and the final answer was −625256-\frac{625}{256}.

Understanding the Basics

In the previous article, we evaluated two expressions with exponents and wrote our answers as fractions. However, we may still have some questions about how to evaluate expressions with exponents. In this article, we will answer some of the most frequently asked questions about evaluating expressions with exponents.

Q&A

Q: What is the order of operations when evaluating expressions with exponents?

A: The order of operations when evaluating expressions with exponents is PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction). This means that we need to follow the order of operations to evaluate expressions with exponents.

Q: What happens when a negative number is raised to an even power?

A: When a negative number is raised to an even power, the result is always positive. This is because the even power cancels out the negative sign.

Q: What happens when a negative number is raised to an odd power?

A: When a negative number is raised to an odd power, the result is always negative. This is because the odd power does not cancel out the negative sign.

Q: How do I simplify an expression with a fraction raised to a power?

A: To simplify an expression with a fraction raised to a power, we need to follow the exponent rule (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}. This means that we need to raise both the numerator and the denominator to the power of n.

Q: How do I simplify an expression with a negative sign in front of a fraction raised to a power?

A: To simplify an expression with a negative sign in front of a fraction raised to a power, we need to follow the order of operations and first deal with the exponent. Then, we can simplify the expression by raising the fraction to the power of n and multiplying it by the negative sign.

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

A: A positive exponent means that we need to raise the base to the power of the exponent, while a negative exponent means that we need to take the reciprocal of the base and raise it to the power of the absolute value of the exponent.

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

A: To evaluate an expression with multiple exponents, we need to follow the order of operations and first deal with the exponents. Then, we can simplify the expression by raising the base to the power of the exponents.

Examples

Example 1: Evaluating an Expression with a Positive Exponent

Evaluate the expression (23)4\left(\frac{2}{3}\right)^4.

Solution

To evaluate this expression, we need to follow the exponent rule (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}. This means that we need to raise both the numerator and the denominator to the power of 4.

(23)4=2434\left(\frac{2}{3}\right)^4 = \frac{2^4}{3^4}

Using the exponent rule am⋅an=am+na^m \cdot a^n = a^{m+n}, we can simplify the expression further.

2434=1681\frac{2^4}{3^4} = \frac{16}{81}

So, the final answer to the expression is 1681\frac{16}{81}.

Example 2: Evaluating an Expression with a Negative Exponent

Evaluate the expression −(34)3-\left(\frac{3}{4}\right)^3.

Solution

To evaluate this expression, we need to follow the order of operations and first deal with the exponent. Then, we can simplify the expression by raising the fraction to the power of 3 and multiplying it by the negative sign.

−(34)3=−3343-\left(\frac{3}{4}\right)^3 = -\frac{3^3}{4^3}

Using the exponent rule am⋅an=am+na^m \cdot a^n = a^{m+n}, we can simplify the expression further.

−3343=−2764-\frac{3^3}{4^3} = -\frac{27}{64}

So, the final answer to the expression is −2764-\frac{27}{64}.

Conclusion

In this article, we answered some of the most frequently asked questions about evaluating expressions with exponents. We covered topics such as the order of operations, simplifying expressions with fractions raised to powers, and evaluating expressions with multiple exponents. We also provided examples to illustrate how to evaluate expressions with exponents. By following the order of operations and simplifying expressions with exponents, we can evaluate complex expressions and find the final answer.