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Chapter 4: Problem 18

Find each product or quotient. Express using exponents. $$a^{6} \cdot a^{6}$$

Short Answer

Expert verified
The product is \(a^{12}\).

Step by step solution

01

Recognize the Operation and Base

The problem asks us to find the product of two terms: \(a^6\) and \(a^6\). Both terms have the same base \(a\).
02

Apply the Law of Exponents for Products

When multiplying powers with the same base, we add the exponents. Thus, \(a^6 \cdot a^6 = a^{6+6}\).
03

Simplify the Exponent

Add the exponents together: \(6 + 6 = 12\).
04

Write the Final Expression

Express the product using exponents: \(a^{12}\).

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Key Concepts

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

Multiplication of Powers
The multiplication of powers is a fundamental concept in algebra that deals with expressions where terms with exponents are multiplied. This concept is particularly useful when you have to multiply terms that have the same base. In algebra, a "base" refers to the repeated factor in an expression involving exponents.

When multiplying powers with the same base, you keep the base the same and add the exponents together. For example:
  • If you have two powers like \( x^m \cdot x^n \), you simply add the exponents \( m \) and \( n \).
  • This gives you the expression \( x^{m+n} \).
The rule to recall here is that only the exponents are added, not the bases themselves. This is because multiplication of numbers is fundamentally grouping repeated additions together, which is mirrored in this exponent rule.
Law of Exponents
The Law of Exponents is a set of rules that describe how to manipulate powers. One of the most commonly used laws is the product of powers property, which states that when you multiply two powers with the same base, you add their exponents.

This law helps simplify complex expressions and makes calculations much easier. For example:
  • \( b^4 \cdot b^3 = b^{4+3} = b^7 \)
  • \( c^5 \cdot c^2 \cdot c^3 = c^{5+2+3} = c^{10} \)
Using these identities allows one to quickly transform and solve expressions involving powers. Other laws, like the zero exponent rule or power of a power, are also essential but the focus here is on how to apply them to multiplication.
Simplifying Expressions
Simplifying expressions is the process of reducing a complex expression into its simplest form. This often involves combining like terms or using mathematical laws, like those of exponents, to make the equation easier to work with or understand.

Let鈥檚 look at simplifying exponents with the power rule:
  • You can often combine several steps into one by directly adding or subtracting exponent values when bases are the same.
  • In our example of multiplying \( a^6 \) with \( a^6 \), recognizing that we have the same base allows us to use our rule to quickly arrive at \( a^{12} \).
By not skipping these simplification steps, not only does the work become cleaner but it also reduces potential errors. Understanding and applying the correct laws is key to mastering simplification.

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Most popular questions from this chapter

Write and solve an equation to find each number. The sum of a number and 9 is \(-2\)

Which One Doesn't Belong? Identify the fraction that does not belong with the other three. Explain your reasoning. $$\frac{6 y}{5}$$ $$\frac{4}{7}$$ $$\frac{1}{x^{2}}$$ $$\frac{15}{12}$$

Use the following information. Musical notes \(C\) and A sound harmonious together because of their frequencies, or vibrations. The fraction that is formed by the two frequencies can be simplified, as shown below. $$ \frac{C}{A}=\frac{264}{440} \text { or } \frac{3}{5} $$ $$ \begin{array}{|c|c|} \hline \text { Note } & \text { Frequency (hz) } \\ \hline \mathrm{C} & 264 \\ \mathrm{D} & 294 \\ \mathrm{E} & 330 \\ \mathrm{~F} & 349 \\ \mathrm{G} & 392 \\ \mathrm{~A} & 440 \\ \mathrm{~B} & 494 \\ \mathrm{C} & 528 \\ \hline \end{array} $$ When a fraction formed by two frequencies cannot be simplified, the notes sound like noise. Determine whether each pair of notes would sound harmonious together. Explain why or why not. $$ \mathrm{E} \text { and } \mathrm{A} $$

Find each product or quotient. Express using exponents. $$\frac{36 d^{6}}{12 d^{4}}$$

For each expression, use parentheses to group the numbers together and to group the powers with like bases together EXAMPLE \(: a \cdot 4 \cdot a^{3} \cdot 2=(4 \cdot 2)\left(a \cdot a^{3}\right)\) \(b \cdot 5 \cdot 10 \cdot b^{4}\)
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