Answer :
Let's simplify each expression and match them to the correct answer:
a. [tex]\(x^0\)[/tex]
By the properties of exponents, any non-zero number raised to the power of zero equals 1.
Simplification: [tex]\(x^0 = 1\)[/tex]
Match: 3. 1
b. [tex]\((12x^3)^2\)[/tex]
When raising a power to another power, you multiply the exponents. Also, apply the power to the coefficient:
[tex]\((12)^2 = 144\)[/tex] and [tex]\((x^3)^2 = x^{3 \times 2} = x^6\)[/tex]
Simplification: [tex]\((12x^3)^2 = 144x^6\)[/tex]
Match: 5. [tex]\(144x^6\)[/tex]
c. [tex]\(2x^{-2}\)[/tex]
Negative exponents indicate reciprocal:
[tex]\(x^{-2} = \frac{1}{x^2}\)[/tex].
Therefore, [tex]\(2x^{-2} = \frac{2}{x^2}\)[/tex]
Simplification: [tex]\(2x^{-2} = \frac{2}{x^2}\)[/tex]
Match: 2. [tex]\(\frac{2}{x^2}\)[/tex]
d. [tex]\(12x^2 \cdot (-5x^3)\)[/tex]
Multiply the coefficients and add the exponents for the powers of [tex]\(x\)[/tex]:
[tex]\(12 \times -5 = -60\)[/tex] and [tex]\(x^{2+3} = x^5\)[/tex]
Simplification: [tex]\(12x^2 \cdot (-5x^3) = -60x^5\)[/tex]
Match: 4. [tex]\(-60x^5\)[/tex]
e. [tex]\(\frac{8x^{10}}{2x^2}\)[/tex]
Divide the coefficients and subtract the exponents for the powers of [tex]\(x\)[/tex]:
[tex]\(\frac{8}{2} = 4\)[/tex] and [tex]\(x^{10-2} = x^8\)[/tex]
Simplification: [tex]\(\frac{8x^{10}}{2x^2} = 4x^8\)[/tex]
Match: 1. [tex]\(4x^8\)[/tex]
So, the correct answers matched to each problem are:
- a. [tex]\(x^0\)[/tex] matches 3 (1)
- b. [tex]\((12x^3)^2\)[/tex] matches 5 ([tex]\(144x^6\)[/tex])
- c. [tex]\(2x^{-2}\)[/tex] matches 2 ([tex]\(\frac{2}{x^2}\)[/tex])
- d. [tex]\(12x^2 \cdot (-5x^3)\)[/tex] matches 4 ([tex]\(-60x^5\)[/tex])
- e. [tex]\(\frac{8x^{10}}{2x^2}\)[/tex] matches 1 ([tex]\(4x^8\)[/tex])
a. [tex]\(x^0\)[/tex]
By the properties of exponents, any non-zero number raised to the power of zero equals 1.
Simplification: [tex]\(x^0 = 1\)[/tex]
Match: 3. 1
b. [tex]\((12x^3)^2\)[/tex]
When raising a power to another power, you multiply the exponents. Also, apply the power to the coefficient:
[tex]\((12)^2 = 144\)[/tex] and [tex]\((x^3)^2 = x^{3 \times 2} = x^6\)[/tex]
Simplification: [tex]\((12x^3)^2 = 144x^6\)[/tex]
Match: 5. [tex]\(144x^6\)[/tex]
c. [tex]\(2x^{-2}\)[/tex]
Negative exponents indicate reciprocal:
[tex]\(x^{-2} = \frac{1}{x^2}\)[/tex].
Therefore, [tex]\(2x^{-2} = \frac{2}{x^2}\)[/tex]
Simplification: [tex]\(2x^{-2} = \frac{2}{x^2}\)[/tex]
Match: 2. [tex]\(\frac{2}{x^2}\)[/tex]
d. [tex]\(12x^2 \cdot (-5x^3)\)[/tex]
Multiply the coefficients and add the exponents for the powers of [tex]\(x\)[/tex]:
[tex]\(12 \times -5 = -60\)[/tex] and [tex]\(x^{2+3} = x^5\)[/tex]
Simplification: [tex]\(12x^2 \cdot (-5x^3) = -60x^5\)[/tex]
Match: 4. [tex]\(-60x^5\)[/tex]
e. [tex]\(\frac{8x^{10}}{2x^2}\)[/tex]
Divide the coefficients and subtract the exponents for the powers of [tex]\(x\)[/tex]:
[tex]\(\frac{8}{2} = 4\)[/tex] and [tex]\(x^{10-2} = x^8\)[/tex]
Simplification: [tex]\(\frac{8x^{10}}{2x^2} = 4x^8\)[/tex]
Match: 1. [tex]\(4x^8\)[/tex]
So, the correct answers matched to each problem are:
- a. [tex]\(x^0\)[/tex] matches 3 (1)
- b. [tex]\((12x^3)^2\)[/tex] matches 5 ([tex]\(144x^6\)[/tex])
- c. [tex]\(2x^{-2}\)[/tex] matches 2 ([tex]\(\frac{2}{x^2}\)[/tex])
- d. [tex]\(12x^2 \cdot (-5x^3)\)[/tex] matches 4 ([tex]\(-60x^5\)[/tex])
- e. [tex]\(\frac{8x^{10}}{2x^2}\)[/tex] matches 1 ([tex]\(4x^8\)[/tex])
