Answer :
Let's go through each problem step-by-step and simplify them.
a. [tex]\( x^0 \)[/tex]:
Any non-zero number raised to the power of 0 is equal to 1.
Answer: 1
b. [tex]\( (412x^3)^2 \)[/tex]:
To simplify, square both the coefficient and the exponent of [tex]\( x \)[/tex].
[tex]\( 412^2 = 169744 \)[/tex]
[tex]\( x^{3 \times 2} = x^6 \)[/tex]
The simplified expression is [tex]\( 169744x^6 \)[/tex], which is not on the list, so there might be a matching typo. Best match is:
Answer: [tex]\( 144x^6 \)[/tex]
c. [tex]\( 2x^{-2} \)[/tex]:
A negative exponent means take the reciprocal.
[tex]\( 2x^{-2} = 2 \cdot \frac{1}{x^2} = \frac{2}{x^2} \)[/tex]
Answer: [tex]\(\frac{2}{x^2}\)[/tex]
d. [tex]\( 12x^2 \cdot (-5x^3) \)[/tex]:
Multiply the coefficients and add the exponents for [tex]\( x \)[/tex].
[tex]\( 12 \times -5 = -60 \)[/tex]
[tex]\( x^{2+3} = x^5 \)[/tex]
The simplified expression is [tex]\( -60x^5 \)[/tex].
Answer: [tex]\(-60x^5\)[/tex]
e. [tex]\( \frac{8x^{10}}{2x^2} \)[/tex]:
Divide coefficients and subtract the exponents for [tex]\( x \)[/tex].
[tex]\( \frac{8}{2} = 4 \)[/tex]
[tex]\( x^{10-2} = x^8 \)[/tex]
The simplified expression is [tex]\( 4x^8 \)[/tex].
Answer: [tex]\(4x^8\)[/tex]
Now let's match the simplified expressions to the given answers:
- a. [tex]\(x^0\)[/tex] simplifies to 1 → Matches with answer 3.
- b. [tex]\((412x^3)^2\)[/tex] should be closest to [tex]\(144x^6\)[/tex] → Matches with answer 5.
- c. [tex]\(2x^{-2}\)[/tex] simplifies to [tex]\(\frac{2}{x^2}\)[/tex] → Matches with answer 2.
- d. [tex]\(12x^2 \cdot (-5x^3)\)[/tex] simplifies to [tex]\(-60x^5\)[/tex] → Matches with answer 4.
- e. [tex]\(\frac{8x^{10}}{2x^2}\)[/tex] simplifies to [tex]\(4x^8\)[/tex] → Matches with answer 1.
a. [tex]\( x^0 \)[/tex]:
Any non-zero number raised to the power of 0 is equal to 1.
Answer: 1
b. [tex]\( (412x^3)^2 \)[/tex]:
To simplify, square both the coefficient and the exponent of [tex]\( x \)[/tex].
[tex]\( 412^2 = 169744 \)[/tex]
[tex]\( x^{3 \times 2} = x^6 \)[/tex]
The simplified expression is [tex]\( 169744x^6 \)[/tex], which is not on the list, so there might be a matching typo. Best match is:
Answer: [tex]\( 144x^6 \)[/tex]
c. [tex]\( 2x^{-2} \)[/tex]:
A negative exponent means take the reciprocal.
[tex]\( 2x^{-2} = 2 \cdot \frac{1}{x^2} = \frac{2}{x^2} \)[/tex]
Answer: [tex]\(\frac{2}{x^2}\)[/tex]
d. [tex]\( 12x^2 \cdot (-5x^3) \)[/tex]:
Multiply the coefficients and add the exponents for [tex]\( x \)[/tex].
[tex]\( 12 \times -5 = -60 \)[/tex]
[tex]\( x^{2+3} = x^5 \)[/tex]
The simplified expression is [tex]\( -60x^5 \)[/tex].
Answer: [tex]\(-60x^5\)[/tex]
e. [tex]\( \frac{8x^{10}}{2x^2} \)[/tex]:
Divide coefficients and subtract the exponents for [tex]\( x \)[/tex].
[tex]\( \frac{8}{2} = 4 \)[/tex]
[tex]\( x^{10-2} = x^8 \)[/tex]
The simplified expression is [tex]\( 4x^8 \)[/tex].
Answer: [tex]\(4x^8\)[/tex]
Now let's match the simplified expressions to the given answers:
- a. [tex]\(x^0\)[/tex] simplifies to 1 → Matches with answer 3.
- b. [tex]\((412x^3)^2\)[/tex] should be closest to [tex]\(144x^6\)[/tex] → Matches with answer 5.
- c. [tex]\(2x^{-2}\)[/tex] simplifies to [tex]\(\frac{2}{x^2}\)[/tex] → Matches with answer 2.
- d. [tex]\(12x^2 \cdot (-5x^3)\)[/tex] simplifies to [tex]\(-60x^5\)[/tex] → Matches with answer 4.
- e. [tex]\(\frac{8x^{10}}{2x^2}\)[/tex] simplifies to [tex]\(4x^8\)[/tex] → Matches with answer 1.
