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Chapter 1: Problem 15

Evaluate each expression without using a calculator. $$ \left[\left(\frac{2}{3}\right)^{-2}\right]^{-1} $$

Short Answer

Expert verified
The expression evaluates to \(\frac{4}{9}\).

Step by step solution

01

Understand Negative Exponents

The expression \(\left(\frac{2}{3}\right)^{-2}\) implies that we take the reciprocal of \(\frac{2}{3}\) and then raise it to the positive exponent. Therefore, \(\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2}\).
02

Apply Exponent to the Fraction

Now we need to calculate \(\left(\frac{3}{2}\right)^{2}\). This involves squaring both the numerator and the denominator: \(\frac{3^2}{2^2} = \frac{9}{4}\).
03

Simplify the Expression with Another Negative Exponent

The entire expression is raised to a negative exponent, \(\left[\left(\frac{2}{3}\right)^{-2}\right]^{-1} = \left(\frac{9}{4}\right)^{-1}\), which requires taking the reciprocal of \(\frac{9}{4}\), resulting in \(\frac{4}{9}\).
04

Verify the Solution

To ensure the calculations are correct, remember that taking the reciprocal twice (due to the double negative exponents) should return the final fraction to \(\frac{4}{9}\). This confirms the operation correctness.

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Key Concepts

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

Negative Exponents
A negative exponent may look confusing at first, but it's actually quite simple. When you see an expression like \( a^{-b} \), it means you need to take the reciprocal of \( a \) and raise it to the positive exponent \( b \). This is because the negative sign indicates an inverse operation.

Here's a breakdown:
  • The expression \( 3^{-2} \) becomes \( \frac{1}{3^2} \).
  • This means \( 3^{-2} = \frac{1}{9} \).
By reinterpreting negative exponents in this way, they become much easier to handle.

So, always remember: a negative exponent flips the base and makes the exponent positive.
Reciprocal
The reciprocal of a number is essentially what you multiply the original number by to get 1. In simple terms, if you have a fraction like \( \frac{a}{b} \), its reciprocal will be \( \frac{b}{a} \).

Here's why this is useful:
  • If you multiply \( \frac{a}{b} \) and \( \frac{b}{a} \) together, you get 1 because \( \frac{a}{b} \times \frac{b}{a} = \frac{ab}{ab} = 1 \).
  • Reciprocals turn division into multiplication, simplifying computations greatly, especially when dealing with fractions and exponents.
Understanding reciprocals can make solving problems with negative exponents more intuitive, as a negative exponent will naturally invert the fraction.
Fraction Exponents
Fraction exponents are another intriguing concept where exponents appear as fractions. They provide a concise way to express roots. For any expression of the form \( a^{\frac{m}{n}} \), it represents the nth root of \( a^m \). The numerator 'm' is the power, and the denominator 'n' is the root.

Consider the following example:
  • \( 8^{\frac{1}{3}} \) translates to the cube root of 8, which equals 2.
  • \( 27^{\frac{2}{3}} \) equates to the cube root of 27 squared, which simplifies to 3 squared, or 9.
Thus, understanding fraction exponents empowers you to transform complex root calculations into a straightforward form.
Exponentiation
Exponentiation is one of the fundamental operations in mathematics, allowing us to express repeated multiplication neatly. When we say \( a^b \), it means multiplying \( a \) by itself \( b \) times. This concept is crucial for tackling problems involving powers and roots.

It generally involves:
  • Multiplying the base repeatedly, as indicated by the power.
  • Handling special cases like negative exponents, which imply reciprocals.
  • Managing fractional exponents, where roots and powers combine.
By mastering exponentiation, you build a solid foundation for understanding more advanced mathematical concepts, such as scientific notation and logarithms.

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