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Chapter 5: Problem 8

Simplify: \(\left(a^{2-n}\right)^{3}\)

Short Answer

Expert verified
The simplified form is \( a^{6-3n} \).

Step by step solution

01

Apply Power of a Power Rule

To simplify an expression in the form \((x^m)^n\), apply the power of a power rule: multiply the exponents \((m \cdot n)\). Here, we have \( (a^{2-n})^{3} \), so we will multiply the exponents \( (2-n) \) and \( 3 \).
02

Simplify the Exponent

Multiply the exponents: \( (2-n) \cdot 3 = 3(2-n) \). Distribute the \(3\) across the \(2-n\): \(3 \cdot 2 - 3 \cdot n = 6 - 3n \).
03

Rewrite the Expression

Substitute the simplified exponent back into the expression: \( a^{6-3n} \). This is the simplified form of the original expression.

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Key Concepts

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

Power of a Power Rule
The power of a power rule is incredibly useful in simplifying expressions with exponents. Think of it as a way to "unlayer" complex exponent scenarios. When you encounter an expression like
  • \( (x^m)^n = x^{m\cdot n} \),
what you're doing is multiplying the exponents, instead of raising a power to a power step-by-step.
Using this rule helps streamline complex algebraic expressions and makes solving them less intimidating.
In the given problem, we've applied the power of a power rule to simplify
  • \( (a^{2-n})^3 \).
By multiplying the exponents, we transform the expression into a single exponent, rather than calculating a cubic or higher power.
Exponents
Exponents are an essential component of algebra, acting as a concise way to express repeated multiplication. They come into play whenever you need to compactly represent multiplying a number by itself a certain number of times. An exponent,
  • shown as the small number to the upper right of a base (e.g., \( a^3 \)),
indicates how many times the base is used as a factor.
For instance, \( a^3 \) means \( a \times a \times a \).Exponents follow specific rules that allow them to be manipulated, such as:
  • Multiplying powers with the same base, where you add the exponents.
  • Raising a power to a power, where you multiply the exponents.
Understanding these rules can simplify even the most convoluted expressions, just like in our exercise.
Algebraic Expressions
Algebraic expressions are at the heart of algebra and are composed of variables, numbers, and operations.
They can represent real-world situations, serve as formulas, or stand alone as abstract entities for manipulation. Here's what you should know about them:
  • A variable is a symbol, often a letter, that represents a number.
  • An algebraic expression includes one or more terms linked together by operations like addition, subtraction, multiplication, etc.
In our problem,
  • the expression \( (a^{2-n})^3 \) is simplified into its most compact form \( a^{6-3n} \).
Through manipulation using rules like the power of a power rule, algebraic expressions can be transformed into more manageable forms.
Thus, grasping how they work helps in performing operations efficiently and accurately.

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Most popular questions from this chapter

Simplify: \((x+5)(x-5)\).

Simplify the following polynomial, and arrange your answer in ascending powers of \(b:\) $$5 a^{3} b^{2}+4 a b^{3}-2 a^{2} b+3 a^{3} b^{2}-a b^{3}$$

\(3^{4} \cdot 3^{2}\)

\(\frac{4^{5}}{4^{3}}\)

Given \(f(x)=2 x^{2}+9 x-5\) and \(g(x)=-x^{2}-4 x+3\), simplify \(f(x)+g(x)\).
See all solutions

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