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Chapter 1: Problem 12

Simplify. $$\left(-3 x^{-2}\right)(4)$$

Short Answer

Expert verified
The simplified expression is \(-\frac{12}{x^2}\).

Step by step solution

01

Distribute the Constants

Multiply the constants from the terms together. Here, we have \(-3\) and \(4\). Multiply them: \(-3 \times 4 = -12\).
02

Simplify the Power of the Variable

The given expression is \(x^{-2}\). Recall that a negative exponent means the reciprocal: \(x^{-2} = \frac{1}{x^2}\).
03

Combine Results

Combine the results from the previous steps. The constant factor is \(-12\) and the variable part is \(x^{-2}\). Therefore, \(-3x^{-2} \times 4 = -12x^{-2} = \frac{-12}{x^2}\).

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Key Concepts

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

Negative Exponents Explained
Understanding negative exponents is crucial in algebraic simplification. A negative exponent indicates that rather than multiplying the base, you actually divide by it. This is because a negative exponent represents the reciprocal of the base raised to the positive exponent.

When you see an expression like \( x^{-2} \), it translates to the reciprocal of \( x \) raised to the power of \( 2 \), or \( \frac{1}{x^2} \). This means that instead of multiplying \( x \) twice, you divide \( 1 \) by \( x^2 \).
  • Negative exponents "flip" the base 鈥� putting it under 1.
  • \( x^{-n} = \frac{1}{x^n} \).
So, when simplifying expressions with negative exponents, remember to transform them into their reciprocal form.
Multiplying Constants in Algebra
In algebra, multiplying constants is a straightforward but crucial step in simplification. Constants are simply numbers that do not change and can be multiplied directly as you would with regular arithmetic.
  • Identify the constants in the expression.
  • Multiply them together.
  • The result becomes the new coefficient of your variable or expression.
For instance, given the problem \((-3x^{-2})(4)\), first identify the constants: \(-3\) and \(4\). Multiplying them gives \(-12\).

This step ensures that your expression is simpler, making it easier to work with or further simplify.
Understanding Reciprocal of Powers
The concept of the reciprocal of powers sheds light on why negative exponents behave as they do. When you have a power, say \(x^n\), its reciprocal is \(\frac{1}{x^n}\). This relationship is directly linked to how negative exponents are defined.

Reciprocals change the position of the base in the fraction. If \(x\) is in the numerator, its reciprocal \(\frac{1}{x}\) positions it in the denominator, and vice versa.
  • Reciprocal of \(x^n\): \(\frac{1}{x^n}\).
  • Turning powers into reciprocals helps to simplify expressions with division.

In algebraic operations, utilizing the reciprocal of powers is particularly useful when dealing with division or negative exponents, ensuring a smoother and more accurate simplification process.

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Most popular questions from this chapter

Simplify the expression. $$\frac{9 x^{2}-4}{3 x^{2}-5 x+2} \cdot \frac{9 x^{4}-6 x^{3}+4 x^{2}}{27 x^{4}+8 x}$$

In evaluating negative numbers raised to fractional powers, it may be necessary to evaluate the root and integer power separately. For example, \((-3)^{2 / 5}\) can be evaluated successfully as \(\left[(-3)^{1 / 5}\right]^{2}\) or \(\left[(-3)^{2}\right]^{1 / 5}\), whereas an error message might otherwise appear. Approximate the realnumber expression to four decimal places. (a) \((-3)^{2 / 5}\) (b) \((-7)^{4 / 3}\)

Simplify the expression. $$\frac{\frac{1}{(x+h)^{3}}-\frac{1}{x^{3}}}{h}$$

Simplify the expression. $$\frac{(6 x+1)^{3}\left(27 x^{2}+2\right)-\left(9 x^{3}+2 x\right)(3)(6 x+1)^{2}(6)}{(6 x+1)^{6}}$$

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[4]{\left(5 x^{5} y^{-2}\right)^{4}}$$
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