Let's solve each part of the question step-by-step and match it to the given multiple choice answers.
Part (a): [tex]\( x^0 \)[/tex]
Any number, including [tex]\( x \)[/tex], raised to the power of 0 is equal to 1 (except when the base is 0). Thus,
[tex]\[ x^0 = 1 \][/tex]
Answer: 1
Part (b): [tex]\(\left(12 x^3\right)^2 \)[/tex]
To simplify this, use the power of a power property: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. This property states that you multiply the exponents.
[tex]\[ \left(12 x^3\right)^2 = 12^2 \cdot (x^3)^2 \][/tex]
[tex]\[ = 144 \cdot x^{3 \cdot 2} \][/tex]
[tex]\[ = 144 x^6 \][/tex]
Answer: 144 [tex]\(x^6\)[/tex]
Part (c): [tex]\(2 x^{-2} \)[/tex]
A negative exponent means we take the reciprocal of the base raised to the positive exponent. Hence, [tex]\( x^{-2} = \frac{1}{x^2} \)[/tex]:
[tex]\[ 2 x^{-2} = 2 \cdot \frac{1}{x^2} \][/tex]
[tex]\[ = \frac{2}{x^2} \][/tex]
Answer: [tex]\(\frac{2}{x^2} \)[/tex]
Part (d): [tex]\(12 x^2 \cdot\left(-5 x^3\right) \)[/tex]
Use the product of powers property: [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]. Multiply the coefficients and add the exponents of [tex]\( x \)[/tex]:
[tex]\[ 12 x^2 \cdot (-5 x^3) = (12 \cdot -5) \cdot x^{2+3} \][/tex]
[tex]\[ = -60 \cdot x^5 \][/tex]
Answer: -60 [tex]\(x^5\)[/tex]
Part (e): [tex]\(\frac{8 x^{10}}{2 x^2} \)[/tex]
Use the quotient of powers property: [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]. Divide the coefficients and subtract the exponents of [tex]\( x \)[/tex]:
[tex]\[ \frac{8 x^{10}}{2 x^2} = \left(\frac{8}{2}\right) \cdot x^{10-2} \][/tex]
[tex]\[ = 4 \cdot x^8 \][/tex]
Answer: 4 [tex]\(x^8\)[/tex]
Matching Multiple Choice Answers
1. [tex]\( 4 x^8 \)[/tex] : Part (e)
2. [tex]\(\frac{2}{x^2} \)[/tex] : Part (c)
3. 1 : Part (a)
4. [tex]\(-60 x^5 \)[/tex] : Part (d)
5. [tex]\( 144 x^6 \)[/tex] : Part (b)
So, the correct answers are:
- Part (a): 1
- Part (b): [tex]\( 144 x^6 \)[/tex]
- Part (c): [tex]\(\frac{2}{x^2} \)[/tex]
- Part (d): [tex]\(-60 x^5 \)[/tex]
- Part (e): [tex]\( 4 x^8 \)[/tex]
These match options 3, 5, 2, 4, and 1, respectively.
