Answer :
Sure! Let's simplify each expression and match them with the correct answers.
a. [tex]\(x^0\)[/tex]
- According to the property of exponents, any non-zero number raised to the power of 0 is 1.
- So, [tex]\(x^0 = 1\)[/tex].
b. [tex]\((12x^3)^2\)[/tex]
- First, apply the power of a power rule: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
- You get [tex]\((12^1 \cdot x^3)^2 = 12^2 \cdot (x^3)^2\)[/tex].
- Thus, [tex]\(12^2 = 144\)[/tex] and [tex]\((x^3)^2 = x^6\)[/tex].
- Combining these, [tex]\((12x^3)^2 = 144x^6\)[/tex].
c. [tex]\(2x^{-2}\)[/tex]
- A negative exponent means we take the reciprocal: [tex]\(x^{-n} = \frac{1}{x^n}\)[/tex].
- So, [tex]\(2x^{-2} = 2 \cdot \frac{1}{x^2} = \frac{2}{x^2}\)[/tex].
d. [tex]\(12x^2 \cdot (-5x^3)\)[/tex]
- Apply the product of powers rule: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
- Multiply the coefficients: [tex]\(12 \cdot -5 = -60\)[/tex].
- Add the exponents of [tex]\(x\)[/tex]: [tex]\(x^2 \cdot x^3 = x^{2+3} = x^5\)[/tex].
- Therefore, [tex]\(12x^2 \cdot (-5x^3) = -60x^5\)[/tex].
e. [tex]\(\frac{8x^{10}}{2x^2}\)[/tex]
- First, divide the coefficients: [tex]\(\frac{8}{2} = 4\)[/tex].
- Apply the quotient of powers rule: [tex]\(\frac{x^m}{x^n} = x^{m-n}\)[/tex].
- So, [tex]\(\frac{x^{10}}{x^2} = x^{10-2} = x^8\)[/tex].
- Therefore, [tex]\(\frac{8x^{10}}{2x^2} = 4x^8\)[/tex].
Finally, match each simplified expression to its corresponding answer:
1. [tex]\(x^0 = 1\)[/tex] matches with 3.
2. [tex]\((12x^3)^2 = 144x^6\)[/tex] matches with 5.
3. [tex]\(2x^{-2} = \frac{2}{x^2}\)[/tex] matches with 2.
4. [tex]\(12x^2 \cdot (-5x^3) = -60x^5\)[/tex] matches with 4.
5. [tex]\(\frac{8x^{10}}{2x^2} = 4x^8\)[/tex] matches with 1.
And there you have the solution!
a. [tex]\(x^0\)[/tex]
- According to the property of exponents, any non-zero number raised to the power of 0 is 1.
- So, [tex]\(x^0 = 1\)[/tex].
b. [tex]\((12x^3)^2\)[/tex]
- First, apply the power of a power rule: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
- You get [tex]\((12^1 \cdot x^3)^2 = 12^2 \cdot (x^3)^2\)[/tex].
- Thus, [tex]\(12^2 = 144\)[/tex] and [tex]\((x^3)^2 = x^6\)[/tex].
- Combining these, [tex]\((12x^3)^2 = 144x^6\)[/tex].
c. [tex]\(2x^{-2}\)[/tex]
- A negative exponent means we take the reciprocal: [tex]\(x^{-n} = \frac{1}{x^n}\)[/tex].
- So, [tex]\(2x^{-2} = 2 \cdot \frac{1}{x^2} = \frac{2}{x^2}\)[/tex].
d. [tex]\(12x^2 \cdot (-5x^3)\)[/tex]
- Apply the product of powers rule: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
- Multiply the coefficients: [tex]\(12 \cdot -5 = -60\)[/tex].
- Add the exponents of [tex]\(x\)[/tex]: [tex]\(x^2 \cdot x^3 = x^{2+3} = x^5\)[/tex].
- Therefore, [tex]\(12x^2 \cdot (-5x^3) = -60x^5\)[/tex].
e. [tex]\(\frac{8x^{10}}{2x^2}\)[/tex]
- First, divide the coefficients: [tex]\(\frac{8}{2} = 4\)[/tex].
- Apply the quotient of powers rule: [tex]\(\frac{x^m}{x^n} = x^{m-n}\)[/tex].
- So, [tex]\(\frac{x^{10}}{x^2} = x^{10-2} = x^8\)[/tex].
- Therefore, [tex]\(\frac{8x^{10}}{2x^2} = 4x^8\)[/tex].
Finally, match each simplified expression to its corresponding answer:
1. [tex]\(x^0 = 1\)[/tex] matches with 3.
2. [tex]\((12x^3)^2 = 144x^6\)[/tex] matches with 5.
3. [tex]\(2x^{-2} = \frac{2}{x^2}\)[/tex] matches with 2.
4. [tex]\(12x^2 \cdot (-5x^3) = -60x^5\)[/tex] matches with 4.
5. [tex]\(\frac{8x^{10}}{2x^2} = 4x^8\)[/tex] matches with 1.
And there you have the solution!
