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Chapter 7: Problem 16

Simplify each of the following as completely as possible. $$\left(n^{5} n^{4}\right)^{3}$$

Short Answer

Expert verified
The simplified form is \(n^{27}\).

Step by step solution

01

Apply the Product of Powers Property

First, simplify the expression inside the parentheses by using the product of powers rule. The product of powers rule states that when multiplying two exponents with the same base, add the exponents: \(n^5 n^4 = n^{5+4} = n^9\).
02

Apply the Power of a Power Property

Next, apply the power of a power rule to simplify the expression. The power of a power rule states that when raising a power to another power, multiply the exponents: \((n^9)^3 = n^{9 \times 3} = n^{27}\).

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Key Concepts

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

Product of Powers Rule
The Product of Powers Rule is a useful tool when multiplying exponential terms with the same base. When we see something like \(n^5 n^4\), both terms share the base \(n\). According to this rule, we keep the base the same and simply add the exponents together. So, \(n^5 n^4\) becomes:
  • Identify the same base (\(n\))
  • Add the exponents together: \(5 + 4\)
This gives us \(n^9\).
Remember: This rule only works when the bases are the same. If the bases are different, you cannot use the Product of Powers Rule. For example, \(n^5 m^4\) cannot be simplified using this rule.
Power of a Power Rule
Once you have simplified the expression inside the parentheses, you might face a situation where you need to raise a power to another power. This is where the Power of a Power Rule comes in handy.
In our example, after using the Product of Powers Rule, we had \((n^9)^3\). According to the Power of a Power Rule, we multiply the exponents together. Here's how it works:
  • Take the base (\(n\))
  • Multiply the exponents: \(9 \times 3\)
This results in \(n^{27}\).
Always make sure to multiply the exponents correctly to avoid any mistakes in simplifying expressions.
Exponentiation
Exponentiation is a fundamental concept in algebra involving numbers raised to a power. An expression like \(n^3\) means the base \(n\) is multiplied by itself three times: \(n \times n \times n\).
It's essential to understand a few key points about exponentiation:
  • **Base and Exponent:** In \(n^3\), \(n\) is the base, and \(3\) is the exponent.
  • **Multiplication Repeatedly:** The exponent tells you how many times to multiply the base by itself.
  • **Rules Apply:** Various rules like Product of Powers and Power of a Power simplify complex expressions.
Mastering these rules lets you handle higher powers effortlessly and simplifies expressions involving multiple steps.

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Most popular questions from this chapter

Convert each number into standard notation. $$5 \times 10^{3}$$

Estimate the answer without actually carrying out the computation and make the most appropriate choice. If you multiply \(2.08 \times 10^{-7}\) by \(5.14 \times 10^{6},\) the result is closest to (a) 10 (b) 100 (c) 1 (d) 0.1 (e) 0.01

Do your computation using scientific notation. $$(0.004)(250)$$

Perform the indicated operations and simplify. $$\frac{a}{3}-\frac{a}{2}+\frac{a}{5}$$

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(2 a+5 y)(2 a-5 y)$$
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