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Question

Difficulty: EasyProperties of Exponents in Algebraic Expressions

Which of the following expressions is equivalent to 12x84x2\frac{12x^8}{4x^2} for all x≠0x \neq 0?

  1. A
    8x48x^4
  2. B
    8x68x^6
  3. C
    3x43x^4
  4. 3x63x^6Answer
  5. E
    3x103x^{10}

Answer

The expression 3x63x^6
To simplify the expression, divide the coefficients first: 124=3\frac{12}{4} = 3. Then, apply the quotient rule of exponents to the variable terms, which states that for any non-zero base, xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}. Subtracting the exponents gives x8−2=x6x^{8-2} = x^6. Combining these results yields the correct expression 3x63x^6.

Step-by-Step Solution

1
Divide the numerical coefficients of the terms.
124=3\frac{12}{4} = 3
When simplifying a fraction with algebraic terms, the coefficients are divided normally.
2
Apply the quotient rule of exponents to simplify the variable terms.
x8x2=x8−2=x6\frac{x^8}{x^2} = x^{8-2} = x^6
According to the quotient rule of exponents, when dividing expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator.
3
Multiply the simplified coefficient and variable results together.
3x63x^6
Combining the divided coefficient and the simplified variable expression gives the final simplified result.

Key Concept

Quotient Rule of Exponents

Alternative Method

Alternatively, you can expand the exponent terms in the numerator and denominator: 12⋅x⋅x⋅x⋅x⋅x⋅x⋅x⋅x4⋅x⋅x\frac{12 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x}{4 \cdot x \cdot x}. Simplifying the coefficients gives 33, and canceling two pairs of xx from both the numerator and denominator leaves six factors of xx in the numerator, which simplifies to 3x63x^6.
Estimated Time:45s
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Topics

Properties of Exponents in Algebraic Expressions

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