Answer :
Sure! Let's simplify each expression step by step and match them to the correct answers:
a. [tex]\(x^0\)[/tex]
- According to the rule of exponents, any non-zero number raised to the power of zero is 1.
- Therefore, [tex]\(x^0 = 1\)[/tex].
- Match: 3
b. [tex]\((12x^3)^2\)[/tex]
- To simplify, apply the power of a product rule: [tex]\((ab)^n = a^n \cdot b^n\)[/tex].
- So, [tex]\((12x^3)^2 = 12^2 \cdot (x^3)^2\)[/tex].
- Calculating further: [tex]\(12^2 = 144\)[/tex] and [tex]\((x^3)^2 = x^{3 \cdot 2} = x^6\)[/tex].
- The expression simplifies to [tex]\(144x^6\)[/tex].
- Match: 5
c. [tex]\(2x^{-2}\)[/tex]
- A negative exponent means taking the reciprocal, so [tex]\(x^{-n} = \frac{1}{x^n}\)[/tex].
- Therefore, [tex]\(2x^{-2} = 2 \cdot \frac{1}{x^2} = \frac{2}{x^2}\)[/tex].
- Match: 2
d. [tex]\(12x^2 \cdot (-5x^3)\)[/tex]
- First, multiply the coefficients: [tex]\(12 \times -5 = -60\)[/tex].
- Then, for the exponents, use the product of powers rule: [tex]\(x^a \cdot x^b = x^{a+b}\)[/tex].
- Here, [tex]\(x^2 \cdot x^3 = x^{2+3} = x^5\)[/tex].
- So, the expression simplifies to [tex]\(-60x^5\)[/tex].
- Match: 4
e. [tex]\(\frac{8x^{10}}{2x^2}\)[/tex]
- Simplify the fraction by dividing the coefficients: [tex]\(\frac{8}{2} = 4\)[/tex].
- For the variables, use the quotient of powers rule: [tex]\(\frac{x^a}{x^b} = x^{a-b}\)[/tex].
- Thus, [tex]\(\frac{x^{10}}{x^2} = x^{10-2} = x^8\)[/tex].
- The expression simplifies to [tex]\(4x^8\)[/tex].
- Match: 1
So, the correct matching would be:
- a. [tex]\(x^0 = 1\)[/tex] matches with 3
- b. [tex]\((12x^3)^2 = 144x^6\)[/tex] matches with 5
- c. [tex]\(2x^{-2} = \frac{2}{x^2}\)[/tex] matches with 2
- d. [tex]\(12x^2 \cdot (-5x^3) = -60x^5\)[/tex] matches with 4
- e. [tex]\(\frac{8x^{10}}{2x^2} = 4x^8\)[/tex] matches with 1
a. [tex]\(x^0\)[/tex]
- According to the rule of exponents, any non-zero number raised to the power of zero is 1.
- Therefore, [tex]\(x^0 = 1\)[/tex].
- Match: 3
b. [tex]\((12x^3)^2\)[/tex]
- To simplify, apply the power of a product rule: [tex]\((ab)^n = a^n \cdot b^n\)[/tex].
- So, [tex]\((12x^3)^2 = 12^2 \cdot (x^3)^2\)[/tex].
- Calculating further: [tex]\(12^2 = 144\)[/tex] and [tex]\((x^3)^2 = x^{3 \cdot 2} = x^6\)[/tex].
- The expression simplifies to [tex]\(144x^6\)[/tex].
- Match: 5
c. [tex]\(2x^{-2}\)[/tex]
- A negative exponent means taking the reciprocal, so [tex]\(x^{-n} = \frac{1}{x^n}\)[/tex].
- Therefore, [tex]\(2x^{-2} = 2 \cdot \frac{1}{x^2} = \frac{2}{x^2}\)[/tex].
- Match: 2
d. [tex]\(12x^2 \cdot (-5x^3)\)[/tex]
- First, multiply the coefficients: [tex]\(12 \times -5 = -60\)[/tex].
- Then, for the exponents, use the product of powers rule: [tex]\(x^a \cdot x^b = x^{a+b}\)[/tex].
- Here, [tex]\(x^2 \cdot x^3 = x^{2+3} = x^5\)[/tex].
- So, the expression simplifies to [tex]\(-60x^5\)[/tex].
- Match: 4
e. [tex]\(\frac{8x^{10}}{2x^2}\)[/tex]
- Simplify the fraction by dividing the coefficients: [tex]\(\frac{8}{2} = 4\)[/tex].
- For the variables, use the quotient of powers rule: [tex]\(\frac{x^a}{x^b} = x^{a-b}\)[/tex].
- Thus, [tex]\(\frac{x^{10}}{x^2} = x^{10-2} = x^8\)[/tex].
- The expression simplifies to [tex]\(4x^8\)[/tex].
- Match: 1
So, the correct matching would be:
- a. [tex]\(x^0 = 1\)[/tex] matches with 3
- b. [tex]\((12x^3)^2 = 144x^6\)[/tex] matches with 5
- c. [tex]\(2x^{-2} = \frac{2}{x^2}\)[/tex] matches with 2
- d. [tex]\(12x^2 \cdot (-5x^3) = -60x^5\)[/tex] matches with 4
- e. [tex]\(\frac{8x^{10}}{2x^2} = 4x^8\)[/tex] matches with 1
