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Chapter 6: Problem 9

Decide whether each statement is true or false. Use expanded form to show either that the statement is true or what the correct statement should be. a. \(\left(2^{3}\right)^{2}=2^{6}\) b. \(\left(3^{0}\right)^{4}=3^{4}\) c. \(\left(10^{-2}\right)^{4}=-10^{8}\) @ d. \(\left(5^{-3}\right)^{-4}=5^{12}\)

Short Answer

Expert verified
a) True, b) False (correct: \(1\)), c) False (correct: \(10^{-8}\)), d) True.

Step by step solution

01

Analyze Statement A

The statement is \( \left(2^{3}\right)^{2}=2^{6} \). To verify, we need to apply the power of a power property of exponents: \( \left(a^{m}\right)^{n} = a^{m \times n} \). Here, \( a = 2 \), \( m = 3 \), and \( n = 2 \). Calculating the right-hand side based on this rule results in \( 2^{3 \times 2} = 2^{6} \). Thus, the statement is true.
02

Analyze Statement B

The statement is \( \left(3^{0}\right)^{4}=3^{4} \). First, we evaluate \( 3^{0} \), which is known to equal 1 because any number raised to the power of zero is 1. Then we calculate \( 1^{4} \), which simplifies to 1. Therefore, \( \left(3^{0}\right)^{4} = 1 \), which is not equal to \( 3^{4} \). The statement is false. The correct statement should be \( \left(3^{0}\right)^{4}=1 \).
03

Analyze Statement C

The statement is \( \left(10^{-2}\right)^{4}=-10^{8} \). Again using the power of a power property for exponents, we start by simplifying the left side. \( \left(10^{-2}\right)^{4} = 10^{-2 \times 4} = 10^{-8} \). This differs from the right side, \( -10^{8} \), because the former is \( \frac{1}{10^{8}} \) while the latter is just a large negative number \( -10^{8} \). Thus, the statement is false and the corrected statement is \( \left(10^{-2}\right)^{4}=10^{-8} \).
04

Analyze Statement D

The statement is \( \left(5^{-3}\right)^{-4}=5^{12} \). Using the power of a power property again, we calculate \( \left(a^{-m}\right)^{-n} = a^{m \times n} \). Therefore, \( \left(5^{-3}\right)^{-4} \) simplifies to \( 5^{3 \times 4} = 5^{12} \), which matches the right-hand side. Thus, the statement is true.

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Key Concepts

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

Power of a Power Property
The power of a power property is vital when dealing with exponents. This rule states that when you raise an exponent to another exponent, you multiply the exponents. It's expressed as \((a^m)^n = a^{m \times n}\).
For example, in the equation \((2^3)^2 = 2^6\), this rule is applied by multiplying 3 and 2. You get \(2^{3 \times 2} = 2^6\).
This property simplifies complex expressions by reducing the number of calculations needed. Always remember, raising to a power means multiplying exponents, not adding them.
Expanded Form
Expanded form helps in visualizing and verifying exponent problems. It involves writing numbers out as they are multiplied.
For example, for \((2^3)^2\), expanded form breaks it down to \(2^3 \times 2^3 = (2 \times 2 \times 2) \times (2 \times 2 \times 2)\). When simplified, this equals \(2^6\).
This method is beneficial for verifying statements or deducing errors as it provides a clear view of the multiplication process.
Zero Exponent Rule
The zero exponent rule is a fascinating concept, stating that any non-zero number raised to the power of 0 is 1. Mathematically, it's denoted as \(a^0 = 1\) (where \(a eq 0\)).
Consider the expression \((3^0)^4 = 3^4\). Calculating \(3^0\) gives 1, and \(1^4\) remains 1, not equal to \(3^4\) which is 81. Hence, the statement \((3^0)^4 = 3^4\) is false.
This rule is crucial for simplifying expressions and avoiding mistakes in calculating powers.
Negative Exponents
Negative exponents can be tricky, but they introduce the concept of reciprocal values. The formula is \(a^{-m} = \frac{1}{a^m}\).
A negative exponent indicates how many times to divide by the base rather than multiply. For instance, consider \((10^{-2})^4\). Applying the power of a power property, we simplify it to \(10^{-8}\) which is equal to \(\frac{1}{10^8}\).
Contrast this with \(-10^8\), which represents a completely different value: a large negative number. Understanding this helps prevent common misconceptions with signs and magnitude.

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Most popular questions from this chapter

On average a person sheds 1 million dead skin cells every 40 minutes. (The World in One Day, 1997, p. 16) a. How many dead skin cells does a person shed in an hour? Write your answer in scientific notation. (hi) b. How many dead skin cells does a person shed in a year? (Assume that there are 365 days in a year.) Write your answer in scientific notation.

Write an equivalent expression in the form \(a \cdot b^{n}\). (hi) a. \(3 x \cdot 5 x^{3}\) b. \(x \cdot x^{5}\) c. \(2 x^{3}+2 x^{3}\) d. \(3.5(x+0.15)^{4} \cdot(x+0.15)^{2}\) e. \(\left(2 x^{3}\right)^{3}\) f. \(\left[3(x+0.05)^{3}\right]^{2}\)

Population density is the number of people per square mile. That is, if the population of a country were spread out evenly across an entire nation, the population density would be the number of people in each square mile. a. In 2004 , the population of Mexico was about \(1.0 \times 10^{8}\). Mexico has a land area of about \(7.6 \times 10^{5}\) square miles. What was the population density of Mexico in 2004 ? (Central Intelligence Agency, www.cia.gov) (a) b. In 2004 , the population of Japan was about \(1.3 \times 10^{8}\). Japan has a land area of about \(1.5 \times 10^{5}\) square miles. What was the population density of Japan in 2004 ? (Central Intelligence Agency, www.cia.gov) c. How did the ponulation densities of Mexico and Japan compare in 2004 ?

Use the properties of exponents to rewrite each expression. Use your calculator to check that your expression is equivalent to the original expression. [" \(\square\) See Calculator Note 6B to learn how to check equivalent expressions. 4] a. \((5)(x)(x)(x)(x)\) @ b. \(3 x^{4} \cdot 5 x^{6}\) c. \(4 x^{7} \cdot 2 x^{3}\) d. \(\left(-2 x^{2}\right)\left(x^{2}+x^{4}\right)\)

Give the starting value and constant multiplier for each sequence. Then find the 7 th term of the sequence. a. \(16,20,25,31.25, \ldots\) (a) b. \(27,18,12,8, \ldots\)
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