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Chapter 0: Problem 73

Simplify. $$9^{3 / 2}$$

Short Answer

Expert verified
Simplified form is 27.

Step by step solution

01

Understand the Exponent

The exponent in the expression is \( \frac{3}{2} \). This can be interpreted as \( 9 \) raised to the power of \( \frac{3}{2} \). The fractional exponent can be re-written as a root and a power: \( 9^{3/2} = (9^{1/2})^3 \).
02

Simplify the Square Root

Calculate the square root of \( 9 \). \( 9^{1/2} = \sqrt{9} = 3 \). Now the expression becomes \((3)^3\).
03

Raise to the Power

Raise \( 3 \) to the power of \( 3 \). \( 3^3 = 3 \times 3 \times 3 = 27 \).

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Key Concepts

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

fractional exponents
Fractional exponents, also known as rational exponents, are a way to express roots and powers in one compact form. For example, the expression \(9^{3/2}\) can be interpreted as raising 9 to a power and taking a root.

In this case, \(9^{3/2}\) can be rewritten as \[ 9^{3/2} = (9^{1/2})^3 \] Here, \( 9^{1/2} \) represents the square root of 9, and the 3 outside the parentheses indicates that we should raise the result to the third power.

Generally, fractional exponents follow the form \( a^{m/n} = \root{n}{a^m} \) This principle bridges the gap between traditional exponents and roots.
square root
Square roots are a fundamental type of root where you find a number which, when multiplied by itself, yields the original number. In our example, to find the square root of 9, we look for a number that satisfies the equation: \[ x^2 = 9 \] Solving this, we get \( x = 3 \) since \( 3^2 = 9 \).

Thus, \( 9^{1/2} = \sqrt{9} = 3 \).

Understanding square roots is crucial because they frequently appear in problems involving fractional exponents. The square root operation simplifies the base number of an exponent before we apply any additional powers.
power rules
Power rules simplify calculations involving exponents and make manipulation of exponential expressions easier. Here are a few key rules relevant to our exercise:
  • Product Rule: \( a^m \times a^n = a^{m+n} \)
  • Quotient Rule: \( a^m / a^n = a^{m-n} \)
  • Power of a Power: \( (a^m)^n = a^{m \times n} \)
  • Power of a Product: \( (ab)^m = a^m \times b^m \)


In our exercise, the Power of a Power Rule was utilized when interpreting \( 9^{3/2} \) as \( (9^{1/2})^3 \). This transformation makes it easier to first find the root and then apply the power, breaking the problem into two simpler steps.

Mastering these rules enables swift simplification and error-free calculations in algebra involving exponents and roots.

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Most popular questions from this chapter

In general, people in our society are marrying at a later age. The median age, \(A(t),\) of women at first marriage can be approximated by the linear function \(A(t)=0.08 t+19.7\) where \(t\) is the number of years after \(1950 .\) Thus, \(A(0)\) is the median age of women at first marriage in 1950 \(A(50)\) is the median age in \(2000,\) and so on. a) Find \(A(0), A(1), A(10), A(30),\) and \(A(50)\) b) What was the median age of women at first marriage in \(2008 ?\) c) Graph \(A(t)\)

Find the domain of each function given below. $$f(x)=\frac{2}{x+3}$$

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Determine the domain of each function. $$f(x)=\sqrt[6]{5-x}$$
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