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Answer :
Let's go through the process of simplifying each expression step-by-step and then match them with the correct answer:
a. [tex]\(x^0\)[/tex]
Any non-zero number raised to the power of zero is always 1. So, [tex]\(x^0 = 1\)[/tex].
b. [tex]\((12x^3)^2\)[/tex]
To simplify this expression, we apply the rule of exponents: [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex].
Therefore, [tex]\((12x^3)^2 = 12^2 \cdot (x^3)^2 = 144 \cdot x^{3 \times 2} = 144x^6\)[/tex].
c. [tex]\(2x^{-2}\)[/tex]
A negative exponent indicates the reciprocal, so [tex]\(x^{-2} = \frac{1}{x^2}\)[/tex].
This means [tex]\(2x^{-2} = 2 \cdot \frac{1}{x^2} = \frac{2}{x^2}\)[/tex].
d. [tex]\(12x^2 \cdot (-5x^3)\)[/tex]
To simplify, multiply the coefficients and add the exponents of like bases.
So, [tex]\(12 \cdot (-5) = -60\)[/tex] and [tex]\(x^2 \cdot x^3 = x^{2+3} = x^5\)[/tex].
This results in [tex]\(-60x^5\)[/tex].
e. [tex]\(\frac{8x^{10}}{2x^2}\)[/tex]
First, simplify the coefficients: [tex]\(\frac{8}{2} = 4\)[/tex].
Then, subtract the exponents of like bases: [tex]\(x^{10-2} = x^8\)[/tex].
This yields [tex]\(4x^8\)[/tex].
Now, let's match each simplified expression with the correct answer:
- a. [tex]\(x^0 = 1\)[/tex] matches with option 3.
- b. [tex]\((12x^3)^2 = 144x^6\)[/tex] matches with option 5.
- c. [tex]\(2x^{-2} = \frac{2}{x^2}\)[/tex] matches with option 2.
- d. [tex]\(12x^2 \cdot (-5x^3) = -60x^5\)[/tex] matches with option 4.
- e. [tex]\(\frac{8x^{10}}{2x^2} = 4x^8\)[/tex] matches with option 1.
Thus, the final matches are:
a. → 3
b. → 5
c. → 2
d. → 4
e. → 1
a. [tex]\(x^0\)[/tex]
Any non-zero number raised to the power of zero is always 1. So, [tex]\(x^0 = 1\)[/tex].
b. [tex]\((12x^3)^2\)[/tex]
To simplify this expression, we apply the rule of exponents: [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex].
Therefore, [tex]\((12x^3)^2 = 12^2 \cdot (x^3)^2 = 144 \cdot x^{3 \times 2} = 144x^6\)[/tex].
c. [tex]\(2x^{-2}\)[/tex]
A negative exponent indicates the reciprocal, so [tex]\(x^{-2} = \frac{1}{x^2}\)[/tex].
This means [tex]\(2x^{-2} = 2 \cdot \frac{1}{x^2} = \frac{2}{x^2}\)[/tex].
d. [tex]\(12x^2 \cdot (-5x^3)\)[/tex]
To simplify, multiply the coefficients and add the exponents of like bases.
So, [tex]\(12 \cdot (-5) = -60\)[/tex] and [tex]\(x^2 \cdot x^3 = x^{2+3} = x^5\)[/tex].
This results in [tex]\(-60x^5\)[/tex].
e. [tex]\(\frac{8x^{10}}{2x^2}\)[/tex]
First, simplify the coefficients: [tex]\(\frac{8}{2} = 4\)[/tex].
Then, subtract the exponents of like bases: [tex]\(x^{10-2} = x^8\)[/tex].
This yields [tex]\(4x^8\)[/tex].
Now, let's match each simplified expression with the correct answer:
- a. [tex]\(x^0 = 1\)[/tex] matches with option 3.
- b. [tex]\((12x^3)^2 = 144x^6\)[/tex] matches with option 5.
- c. [tex]\(2x^{-2} = \frac{2}{x^2}\)[/tex] matches with option 2.
- d. [tex]\(12x^2 \cdot (-5x^3) = -60x^5\)[/tex] matches with option 4.
- e. [tex]\(\frac{8x^{10}}{2x^2} = 4x^8\)[/tex] matches with option 1.
Thus, the final matches are:
a. → 3
b. → 5
c. → 2
d. → 4
e. → 1
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