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Chapter 6: Problem 153

Simplify each expression. $$\left(-2 a^{3} b^{2}\right)^{4}$$

Short Answer

Expert verified
16 a^{12} b^{8}

Step by step solution

01

Identify the Expression

The given expression is \((-2 a^{3} b^{2})^{4}\). This involves raising a product of terms to a power.
02

Apply the Power to Each Term

Use the power rule \( (xy)^n = x^n y^n \) to distribute the exponent 4 to each factor inside the parentheses: \((-2)^{4} (a^{3})^{4} (b^{2})^{4}\).
03

Calculate the Constant Term

Simplify \((-2)^{4}\). Since \(-2\) raised to the power of 4 is 16, we get \((-2)^{4} = 16\).
04

Apply the Power to Each Variable

Apply the power rule to both variables: \((a^{3})^{4} = a^{3 \times 4} = a^{12}\) and \((b^{2})^{4} = b^{2 \times 4} = b^{8}\).
05

Combine the Results

Combine the simplified terms to get \(16 a^{12} b^{8}\).

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Key Concepts

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

Power Rule
The Power Rule is a fundamental concept in algebra that can simplify expressions by distributing exponents.
When you have a term like \(x^a\) raised to another power \(b\), the rule states that you multiply the exponents: \((x^a)^b = x^{a*b}\).
For instance, if you have \((a^3)^4\), then using the Power Rule, it becomes \(a^{3*4} = a^{12}\).
This principle is essential in simplifying algebraic expressions where variables are raised to powers within parentheses.
Polynomials
A polynomial is an algebraic expression composed of variables and coefficients, organized in terms of powers.
Each individual term of a polynomial can include constants, variable exponents, or both.
For example, in the expression \(3x^2 + 2x - 5\), there are three terms: \(3x^2\), \(2x\), and \(-5\).
When simplifying polynomials, each term must be considered separately, especially when variables are raised to different powers.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form.
This often includes combining like terms, factoring, and applying rules like the Power Rule.
For example, to simplify \((2x^2 + 3x)(x - 5)\), you would expand the expression and then combine like terms.
Simplification helps in making complex algebraic expressions more manageable, easier to interpret, and solve.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation like addition or multiplication.
They do not contain an equals sign, which differentiates them from equations.
For example, \(4a + 5b - 3\) is an algebraic expression, while \(4a + 5b - 3 = 0\) is an equation.
Simplifying algebraic expressions frequently involves using rules such as the Power Rule and handling various arithmetic operations efficiently.

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Most popular questions from this chapter

Divide each polynomial by the monomial. $$\frac{30 b+75}{5}$$

When Drake simplified \(-3^{0}\) and \((-3)^{0}\) he got the same answer. Explain how using the Order of Operations correctly gives different answers.

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