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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

The population of a town is currently 45,647 . It has been growing at a rate of about \(2.8 \%\) per year. a. Write an expression in the form \(45,647(1+0.028)^{x}\) for the current population. (a) b. What does the expression \(45,647(1+0.028)^{-12}\) represent in this situation? c. Write and evaluate an expression for the population 8 years ago. (a) d. Write expressions without negative exponents that are equivalent to the exponential expressions from \(4 \mathrm{~b}\) and c. (a)

Because the number of molecules in a given amount of a compound is usually a very large number, scientists often work with a quantity called a mole. One mole is about \(6.02 \times 10^{23}\) molecules. a. A liter of water has about \(55.5\) moles of \(\mathrm{H}_{2} \mathrm{O}\). How many molecules is this? Write your answer in scientific notation. b. How many molecules are in \(6.02 \times 10^{23}\) moles of a compound? Write your answer in scientific notation.

Match expressions from this list that are equivalent but written in different exponential forms. There can be multiple matches. a. \(\left(4 x^{4}\right)(3 x)\) b. \(\left(8 x^{2}\right)\left(3 x^{2}\right)\) c. \((12 x)(4 x)\) d. \(\left(6 x^{3}\right)\left(2 x^{2}\right)\) e. \(12 x^{6}\) f. \(24 x^{4}\) g. \(12 x^{5}\) h. \(48 x^{2}\)

Use the distributive property to rewrite each expression in an equivalent form. For example, you can write \(500(1+0.05)\) as \(500+500(0.05)\). a. \(75+75(0.02)\) b. \(1000-1000(0.18)\) (a) c. \(P+P r\) (a) d. \(75(1-0.02)\) e. \(80(1-0.24)\) f. \(A(1-r)\)

A credit card account is essentially a loan. A constant percent interest is added to the balance. Stanley buys \(\$ 100\) worth of groceries with his credit card. The balance then grows by \(1.75 \%\) interest each month. How much will he owe if he makes no payments in 4 months? Write the expression you used to do this calculation in expanded form and also in exponential form. (a)
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