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Chapter 1: Problem 12

Use the laws of exponents to simplify the expressions. $$9^{1 / 3} \cdot 9^{1 / 6}$$

Short Answer

Expert verified
The simplified expression is 3.

Step by step solution

01

Apply the Product of Powers Property

The product of powers property tells us that when multiplying two exponents with the same base, we add the exponents. Here, the base is 9. So we write the expression as:\[ 9^{1/3} \cdot 9^{1/6} = 9^{(1/3 + 1/6)} \]
02

Find a Common Denominator for the Exponents

To add the exponents \( \frac{1}{3} \) and \( \frac{1}{6} \), we need a common denominator. The least common multiple of 3 and 6 is 6. So, we rewrite \( \frac{1}{3} \) as \( \frac{2}{6} \).
03

Add the Exponents

Now, add the exponents using the common denominator:\[ \frac{2}{6} + \frac{1}{6} = \frac{3}{6} \]Simplify \( \frac{3}{6} \) to \( \frac{1}{2} \).
04

Simplify the Expression

Substitute the simplified exponent back into the expression:\[ 9^{(1/3 + 1/6)} = 9^{1/2} \]
05

Express in Simplest Radical Form

Expressing \( 9^{1/2} \) in radical form gives:\[ \sqrt{9} \]Since the square root of 9 is 3, the expression simplifies to:\[ 3 \]

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Key Concepts

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

Product of Powers
When dealing with exponents, the *Product of Powers* property is a useful tool. This property tells us that when we multiply two exponents with the same base, we simply add their exponents together. The base in our example is 9. Thus, the expression \(9^{1/3} \cdot 9^{1/6}\) simplifies using this property:
  • Your exponents are \(\frac{1}{3}\) and \(\frac{1}{6}\).
  • Add them: \(1/3 + 1/6\).
By understanding how to use the product of powers, you can simplify expressions by combining like exponents into one by performing simple addition.
Common Denominator
In order to add fractions, which is a crucial step when using the product of powers with fractions, we need a *common denominator*. For the fractions \(\frac{1}{3}\) and \(\frac{1}{6}\), we look for the smallest common multiple of 3 and 6. This is 6.
  • Convert \(\frac{1}{3}\) into \(\frac{2}{6}\) by multiplying the numerator and denominator by 2.
  • Now you have fractions with the same denominator: \(\frac{2}{6}\) and \(\frac{1}{6}\).
Once you have the fractions with a common denominator, you can easily add them and simplify if possible. In this case, \(\frac{2}{6} + \frac{1}{6} = \frac{3}{6}\), which reduces to \(\frac{1}{2}\).
Simplifying Radicals
Once you've simplified the exponents, it's time to express them in their simplest form - often a radical. Radicals are numbers expressed as roots. Here, we found that \(9^{1/2}\) can be rewritten as a radical.
  • To convert \(9^{1/2}\), recognize that it is the same as \(\sqrt{9}\).
  • The square root of 9 is 3, so \(\sqrt{9} = 3\).
Turning expressions with exponents into their radical form often helps to better understand and visualize the result. Practicing simplifying such expressions ensures dealing with both radicals and exponents becomes intuitive.

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