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Chapter 5: Problem 158

Simplify each expression. \(\left(\frac{k^{-2} k^{8}}{k^{3}}\right)^{2}\)

Short Answer

Expert verified
The simplified expression is \( k^{6} \).

Step by step solution

01

Simplify the expression inside the parentheses

First, simplify \ \( \frac{k^{-2} k^{8}}{k^{3}} \). Use the properties of exponents: \( k^{-2} \cdot k^{8} = k^{-2+8} = k^{6} \). Therefore, the expression becomes \( \frac{k^{6}}{k^{3}} \).
02

Simplify the fraction

Next, apply the exponent rule \( \frac{a^{m}}{a^{n}} = a^{m-n} \) to \( \frac{k^{6}}{k^{3}} \). This gives \( k^{6-3} = k^{3} \). Therefore, the fraction simplifies to \( k^{3} \).
03

Apply the outer exponent

Finally, apply the exponent outside the parentheses, which is 2. This means we need to simplify \( (k^{3})^{2} \). Use the power of a power rule \( (a^{m})^{n} = a^{m \cdot n} \), giving \( (k^{3})^{2} = k^{3 \cdot 2} = k^{6} \).

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Key Concepts

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

Properties of Exponents
Understanding the properties of exponents is essential for simplifying exponential expressions. These properties help in manipulating and combining terms effectively. Here are some key properties:
  • Product of Powers: When multiplying two exponential expressions with the same base, you add the exponents. For example, \(a^{m} \times a^{n} = a^{m+n}\).
  • Quotient of Powers: When dividing two exponential expressions with the same base, you subtract the exponents. For instance, \(\frac{a^{m}}{a^{n}} = a^{m-n}\).
  • Power of a Power: When raising an exponential expression to a power, you multiply the exponents. As in \((a^{m})^{n} = a^{m \times n}\).
Grasping these rules makes simplifying expressions like \(\left(\frac{k^{-2} k^{8}}{k^{3}}\right)^{2}\) more manageable.
Exponent Rule
The exponent rule helps simplify expressions where the base remains consistent, but the exponents change.
Let's break down how this works.
  • In our example, we start with \(\frac{k^{-2} k^{8}}{k^{3}}\). Using the product of powers rule, combine \(k^{-2}\) and \(k^{8}\). This results in \(k^{-2+8} = k^{6} \).
  • Next, apply the quotient of powers rule to \(\frac{k^{6}}{k^{3}}\). This simplifies to \(k^{6-3} = k^{3}\).
Applying these rules step-by-step ensures accuracy and ease in solving complex exponential expressions.
Power of a Power Rule
The power of a power rule is crucial when an expression is raised to another exponent. It states that \((a^{m})^{n} = a^{m \times n}\).
This rule simplifies the final steps of our example:
  • After simplifying the inside expression, we have \((k^{3})^{2}\).
  • Now, apply the power of a power rule. Multiply the exponents: \(k^{3 \times 2} = k^{6}\).
Understanding and applying this rule ensures that you accurately simplify any complex exponential expression efficiently.

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Most popular questions from this chapter

Which method do you prefer to use when multiplying two binomials: the Distributive Property or the FOIL method? Why? Which method do you prefer to use when multiplying a polynomial by a polynomial: the Distributive Property or the Vertical Method? Why?

Multiply the binomials. Use any method. $$ (2 t-9)(10 t+1) $$

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ \left(m+\frac{2}{3} n\right)\left(m-\frac{2}{3} n\right) $$

Square each binomial using the Binomial Squares Pattern. $$ (2 y-3 z)^{2} $$

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ (8 j+4)(8 j-4) $$
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