Match each algebraic expression containing numerical coefficients on the left with its ful… — Armed Services Vocational Aptitude Battery (ASVAB)

Match each algebraic expression containing numerical coefficients on the left with its fully simplified equivalent form on the right.

$3y^2 \cdot 2y^4$ $6y^6$
$\frac{12y^7}{4y^3}$ $3y^4$
$(2y^3)^3$ $8y^9$
$5y^{-2}$ $\frac{5}{y^2}$

Answer

The simplified pairs are:

3y^2 \cdot 2y^4

matches with

6y^6

\frac{12y^7}{4y^3}

matches with

3y^4

(2y^3)^3

matches with

8y^9

, and

5y^{-2}

matches with

\frac{5}{y^2}

Each left-side expression simplifies to exactly one unique form on the right by appropriately applying exponent rules to the variable bases and standard arithmetic operations to the numerical coefficients.

Step-by-Step Solution

Simplify the product expression

3y^2 \cdot 2y^4

6y^6

When multiplying terms, handle coefficients and variables separately. Multiply coefficients (

3 \times 2 = 6

) and add exponents for identical bases (

y^{2+4} = y^6

Simplify the quotient expression

\frac{12y^7}{4y^3}

3y^4

Divide the coefficients (

12 \div 4 = 3

) and subtract the bottom exponent from the top exponent (

y^{7-3} = y^4

Simplify the power expression

(2y^3)^3

8y^9

Raise both the coefficient and the variable to the power of 3. Calculate

2^3 = 8

, and multiply the exponents for the variable

3 \times 3 = 9

Simplify the negative exponent expression

5y^{-2}

\frac{5}{y^2}

A negative exponent creates a reciprocal. Since the exponent only applies to

y

, it moves to the denominator while the coefficient 5 stays in the numerator.

Key Concept

Handling numerical coefficients alongside basic operations with variable exponents.

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