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Chapter 0: Problem 22

Compute the numbers. $$9^{1.5}$$

Short Answer

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Step by step solution

01

Understand the Problem

The task is to compute the value of the given expression which is an exponentiation problem, specifically calculating the value of \(9^{1.5}\).
02

Rewrite the Exponent in Fraction Form

Rewrite the exponent 1.5 as a fraction. Remember that 1.5 can be written as \(\frac{3}{2}\). Thus, \(9^{1.5}\) can be rewritten as \(9^{\frac{3}{2}}\).
03

Simplify the Expression

The expression \(9^{\frac{3}{2}}\) can be broken down into two steps: taking the square root of 9 and then taking the result to the power of 3. Mathematically, this is represented as \((9^{\frac{1}{2}})^3\).
04

Calculate the Square Root

Find the square root of 9. The square root of 9 is 3, so \(9^{\frac{1}{2}} = 3\).
05

Raise to the Power of 3

Raise the result from the previous step to the power of 3. Thus, \(3^3\) which is calculated as 3 * 3 * 3 = 27.

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Key Concepts

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

Fractional Exponents
Fractional exponents can seem tricky at first, but they're just a simpler way to write roots and powers. When you see an exponent that's a fraction, like \(a^{\frac{m}{n}}\), it's a signal to break the problem into two parts: a root and a power.
For example, \(9^{1.5}\) is the same as \(9^{\frac{3}{2}}\).
Here, the numerator (3) tells you the power, and the denominator (2) tells you the root.
Whenever you deal with fractional exponents:
  • Rewrite the fraction in a simpler form if needed.
  • Separate the root and the power.
Square Root
Taking the square root is a fundamental concept in mathematics.
The square root of a number \(a\) is a value \(b\) such that \(b^2 = a\). For instance, \(\sqrt{9} = 3\) because \(3^2 = 9\).
Square roots are essential when simplifying expressions involving fractional exponents.
Consider \(9^{\frac{3}{2}}\): the denominator 2 tells you to find the square root of 9 first.
This makes it easier:
  • Calculate \(\sqrt{9} = 3\).
  • Use the result to finish solving the problem.
Simplification
Simplification is all about making expressions easier to work with.
When dealing with fractional exponents:
  • Break the problem into small, manageable steps.
  • Compute each step carefully.
For instance, solving \(9^{\frac{3}{2}}\) involves first finding the square root of 9: \(9^{\frac{1}{2}} = 3\).
Then, you raise the result to the power of three: \(3^3 = 27\).
The key is staying organized and processing each part in sequence.

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