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Chapter 5: Problem 9

Expand the powers of 10 or \(\frac{1}{10}\). $$\left(\frac{1}{10}\right)^{4}$$

Short Answer

Expert verified
\(\frac{1}{10000}\)

Step by step solution

01

Write the base as a power of 10

The base of the given expression is \ \(\frac{1}{10}\).
02

Apply the power to the base

Raise the fraction \(\frac{1}{10}\) to the power of 4: \ \(\left(\frac{1}{10}\right)^4\).
03

Use the power of a fraction rule

Recall the exponent rule for fractions: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). Therefore, \(\left(\frac{1}{10}\right)^4 = \frac{1^4}{10^4}\).
04

Calculate the powers

Calculate \(1^4\) and \(10^4\): \(1^4 = 1\) and \(10^4 = 10000\). So, \(\frac{1}{10^4} = \frac{1}{10000}\).

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Key Concepts

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

Fractions Exponent Rules
Understanding exponent rules especially when applied to fractions can be very beneficial. One key rule is that when you raise a fraction to a power, both the numerator and the denominator must be raised to that power. This is known as the 'power of a fraction rule'. For a fraction \(\frac{a}{b}\), raised to power 'n', the transformation looks like this:
\[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\ \]
  • If 'a' and 'b' are integers, ensure all calculations adhere to basic arithmetic operations
  • This rule is consistent whether the fraction is simple or complex.
Fractional Exponents
Fractional exponents are another interesting area. A fractional exponent like \(a^{\frac{m}{n}}\) combines both roots and powers. It can be interpreted as:
\[ a^{\frac{m}{n}} = \(\sqrt[n]{a}\)^m = \sqrt[n]{a^m}\ \]
This allows you to understand how to simplify expressions by using the properties of both powers and roots.
  • When you see an exponent as a fraction, remember that the denominator indicates the root, while the numerator indicates the power.
  • For example, \(8^{\frac{2}{3}}\) means taking the cube root of 8 and then raising the result to the power of 2.
Basic Arithmetic Operations
Basic arithmetic operations are the foundation of all math problems, including those involving exponents. These operations include addition, subtraction, multiplication, and division.
The importance of these operations cannot be overstated:
  • Ensure that you follow the order of operations, also known as PEMDAS/BODMAS - Parentheses/Brackets, Orders (i.e., powers and roots, etc.), Multiplication, Division (left to right), Addition, and Subtraction (left to right).
  • Breaking down each arithmetic operation step-by-step is crucial, especially in complex problems. For instance, in our exercise, first handle the powers before the division.
Understanding the fundamental operations ensures accuracy in solving all types of mathematical problems, including those involving exponents and fractions.

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Most popular questions from this chapter

Marty bought a home for \(\$ 145,000 .\) IIe paid \(\$ 25,000\) as a down payment and then financed the rest with a 30 -yr mortgage. His monthly payments are \(\$ 798.36\) and go toward paying off the loan and interest on the loan. a. How much money does Marty have to finance? b. How many months are in a 30 -yr period? c. How much money will Marty pay over a 30 -yr period to pay off the loan? d. How much money did Marty pay in interest over the 30 -yr period?

Perform the indicated operations. $$0 . \overline{5} \div 0 . \overline{2}-0.75$$

Divide. Round the answer to the indicated place value. Use the rounded quotient to check. $$27.88 \div 2.46 \text { thousandths }$$

For Exercises \(71-82,\) simplify by using the order of operations. $$4.2 \div 2.1-(3.1)^{2}$$

The batting averages for five legends are given in the table. Rank the players' batting averages from lowest to highest. (Source: Baseball Almanac) $$\begin{array}{|l|l|} \hline \text { Player } & \text { Average } \\ \hline \text { Joe Jackson } & 0.3558 \\ \hline \text { Ty Cobb } & 0.3664 \\ \hline \text { Lefty O'Doul } & 0.3493 \\ \hline \text { Ted Williams } & 0.3444 \\ \hline \text { Rogers Hornsby } & 0.3585 \\ \hline \end{array}$$
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