Problem 48 Rewrite with a positive exponent... [FREE SOLUTION]

Chapter 10: Problem 48

Rewrite with a positive exponent and evaluate. \(\left(\frac{81}{16}\right)^{-3 / 4}\)

Short Answer

Expert verified

Rewriting the given expression with a positive exponent and evaluating, we get: \(\left(\frac{81}{16}\right)^{-3 / 4} = \left(\frac{16}{81}\right)^{3 / 4} = \left(\frac{4^2}{9^2}\right)^{3 / 4} = \frac{(4^2)^{3 / 4}}{(9^2)^{3 / 4}} = \frac{4^{3/2}}{9^{3/2}} = \left(\frac{4}{9}\right)^{3/2} = \sqrt[2]{\frac{64}{729}} = \frac{8}{27}\)

Step by step solution

Use the property of negative exponents to rewrite the expression with a positive exponent

We have \(\left(\frac{81}{16}\right)^{-3 / 4}\). Applying the property of negative exponents, we get: \(\left(\frac{81}{16}\right)^{-3 / 4} = \left(\frac{16}{81}\right)^{3 / 4}\)

Simplify the expression inside the parentheses

We know that \(81=9^2\) and \(16=4^2\). So we can rewrite the expression inside the parentheses as: \(\frac{16}{81} = \frac{4^2}{9^2}\) Now our expression looks like this: \(\left(\frac{4^2}{9^2}\right)^{3 / 4}\)

Apply the power of a power rule

The power of a power rule states that \((a^n)^m = a^{n \cdot m}\). We can apply this rule to our expression: \(\left(\frac{4^2}{9^2}\right)^{3 / 4} = \frac{(4^2)^{3 / 4}}{(9^2)^{3 / 4}}\)

Evaluate the exponents

Now, we can evaluate the exponents in the numerator and denominator: \((4^2)^{3 / 4} = 4^{(2 \cdot 3 / 4)} = 4^{3/2}\) \((9^2)^{3 / 4} = 9^{(2 \cdot 3 / 4)} = 9^{3/2}\) Now, we can rewrite the expression as \(\frac{4^{3/2}}{9^{3/2}}\)

Apply the quotient rule

The quotient rule states that \(\frac{a^n}{a^m} = a^{n - m}\). Applying the rule to our expression: \(\frac{4^{3/2}}{9^{3/2}} = \left(\frac{4}{9}\right)^{3/2}\)

Evaluate the expression

Finally, we can evaluate the expression: \(\left(\frac{4}{9}\right)^{3/2} = \sqrt[2]{\left(\frac{4}{9}\right)^3} = \sqrt[2]{\frac{64}{729}}\) Since the numerator and denominator are perfect squares, we can take the square root of both: \(\sqrt[2]{\frac{64}{729}} = \frac{\sqrt[2]{64}}{\sqrt[2]{729}} = \frac{8}{27}\) The expression is now rewritten with a positive exponent and evaluated: \(\boxed{\frac{8}{27}}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power of a Power Rule

When dealing with exponents, the power of a power rule is very handy. Imagine you have an expression

ua number raised to another power, such as \((a^n)^m\).
Ithe power of a power rule tells us this is equivalent to raising the base to the product of the exponents, or \(a^{n \cdot m}\).

To apply it, just multiply the exponents together. For example, \( (4^2)^{3/4} \) becomes \( 4^{2 \cdot 3/4} = 4^{3/2} \).
This method simplifies calculations and is crucial when handling nested exponents. By understanding this rule, you can unlock more complex algebraic expressions with accuracy.

Evaluating Expressions

Evaluating expressions involves finding a numerical value for an algebraic expression. Here, it's all about substituting and simplifying:

Start by converting any negative exponents to positive ones.
Substitute the values and apply arithmetic operations carefully.

Consider \(\left(\frac{4}{9}\right)^{3/2}\). Here, you raise the fraction to a power, which breaks down into two steps:

Cubing the fraction: \(\frac{4^3}{9^3}\)
Then taking the square root: \(\sqrt{\frac{64}{729}}\).

Evaluating means turning the expression into an actual number, like \(\frac{8}{27}\), making calculations easier and expressions more understandable.

Simplifying Expressions

Simplifying expressions requires breaking expressions down into simpler forms. This involves:

Combining like terms or using algebraic rules.
Converting complex fractions to more straightforward fractions.

For instance, \(\frac{16}{81} = \frac{4^2}{9^2}\) simplifies this fraction. This simplification helps us see the relationship between the numbers more clearly, making the process of solving much easier.
In the given expression, simplifying reduces steps and allows us to assess the problem efficiently, preparing it for further operations like exponentiation.

Quotient Rule

The quotient rule is a fundamental concept when working with fractions involving powers. It states that when you divide two powers with the same base, you subtract the exponents:

\(\frac{a^n}{a^m} = a^{n-m}\).
This rule can simplify expressions significantly.

In our exercise, however, applying it to different bases leads us to an understanding that the relationship involves entire expressions, such as \(\left(\frac{4}{9}\right)^{3/2}\).
The quotient rule simplifies managing ratios of expressions and is a key step in solving complex algebraic problems efficiently.

91影视

Rewrite with a positive exponent and evaluate. \(\left(\frac{81}{16}\right)^{-3 / 4}\)

Short Answer

Step by step solution

Use the property of negative exponents to rewrite the expression with a positive exponent

Simplify the expression inside the parentheses

Apply the power of a power rule

Evaluate the exponents

Apply the quotient rule

Evaluate the expression

Key Concepts

Power of a Power Rule

Evaluating Expressions

Simplifying Expressions

Quotient Rule

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