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Chapter 0: Problem 32

Use the laws of exponents to compute the numbers. \(\left(9^{4 / 5}\right)^{5 / 8}\)

Short Answer

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Step by step solution

01

Identify the Given Expression

The given expression is \(\left(9^{4 / 5}\right)^{5 / 8}\).
02

Apply the Power Rule for Exponents

Use the power rule of exponents \(\left(a^m\right)^n = a^{m \cdot n}\) to simplify the expression. Here, it becomes \(9^{\left(\frac{4}{5} \cdot \frac{5}{8}\right)}\).
03

Simplify the Exponents

Simplify the exponent \(\frac{4}{5} \cdot \frac{5}{8}\) by multiplying the fractions: \(\frac{4 \cdot 5}{5 \cdot 8} = \frac{20}{40} = \frac{1}{2}\).
04

Rewrite the Expression

Rewrite the simplified expression using the simplified exponent: \(9^{1/2}\).
05

Simplify the Expression Further

Recall that \(9^{1/2}\) is the same as the square root of 9, which is 3.

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Key Concepts

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

Power Rule for Exponents
Let's begin with understanding the power rule for exponents.
This rule states that when you have an exponent raised to another exponent, you can multiply the exponents together.
Mathematically, this is expressed as \((a^m)^n = a^{m \cdot n}\). For example, if you have an expression like \((x^2)^3\), it simplifies to \[x^{2 \cdot 3} = x^6\].
In our exercise, we had \((9^{4/5})^{5/8}\). Here,
we used the power rule to combine the exponents: \(9^{(4/5 \cdot 5/8)}\).
This important step transforms complex expressions into manageable ones.
Simplifying Exponents
Next, let's focus on simplifying exponents.
After applying the power rule for exponents,
we obtained \(9^{(4/5 \cdot 5/8)}\).
To simplify this, we need to multiply the fractions.
Multiplying fractions involves multiplying the numerators together and the denominators together.
So \[ \frac{4}{5} \cdot \frac{5}{8} = \frac{4 \cdot 5}{5 \cdot 8} = \frac{20}{40} = \frac{1}{2} \].
This step is crucial as it reduces the expression to a simpler form: \(9^{1/2}\).
Simplifying exponents is a key technique for breaking down and understanding more complex expressions.
Multiplying Fractions
Multiplying fractions is straightforward once you understand the process.
You simply multiply the numerators (top numbers) together and the denominators (bottom numbers) together.
For our specific example in the exercise: \(\frac{4}{5} \cdot \frac{5}{8}\),
you would do the following:
  • Multiply the numerators: 4 \cdot 5 = 20
  • Multiply the denominators: 5 \cdot 8 = 40
So, \(\frac{4}{5} \cdot \frac{5}{8} = \frac{20}{40}\).
Then, simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD),
which in this case is 20:
\(\frac{20}{40} = \frac{1}{2}\). This gives us the simplified fraction.

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