Answer :
We begin by simplifying each expression one by one.
1. For the expression
[tex]$$x^0,$$[/tex]
recall that any nonzero number raised to the zeroth power is 1. Thus,
[tex]$$x^0 = 1,$$[/tex]
which corresponds to option 3.
2. Next, consider
[tex]$$\left(12x^3\right)^2.$$[/tex]
Square the coefficient and the variable separately:
[tex]$$12^2 = 144,$$[/tex]
and
[tex]$$(x^3)^2 = x^{3 \cdot 2} = x^6.$$[/tex]
Therefore, the expression simplifies to
[tex]$$144x^6,$$[/tex]
which corresponds to option 5.
3. Now simplify
[tex]$$2x^{-2}.$$[/tex]
The negative exponent means we take the reciprocal of the base:
[tex]$$x^{-2} = \frac{1}{x^2}.$$[/tex]
So,
[tex]$$2x^{-2} = \frac{2}{x^2},$$[/tex]
which is option 2.
4. Consider the product
[tex]$$12x^2 \cdot (-5x^3).$$[/tex]
Multiply the coefficients:
[tex]$$12 \times (-5) = -60,$$[/tex]
and add the exponents on [tex]$x$[/tex]:
[tex]$$x^2 \cdot x^3 = x^{2+3} = x^5.$$[/tex]
Hence, the expression simplifies to
[tex]$$-60x^5,$$[/tex]
which corresponds to option 4.
5. Finally, simplify the quotient
[tex]$$\frac{8x^{10}}{2x^2}.$$[/tex]
Divide the coefficients:
[tex]$$\frac{8}{2} = 4,$$[/tex]
and subtract the exponents on [tex]$x$[/tex]:
[tex]$$x^{10} \div x^2 = x^{10-2} = x^8.$$[/tex]
Thus, the expression simplifies to
[tex]$$4x^8,$$[/tex]
which corresponds to option 1.
In summary, the simplified expressions and their matching options are:
- a. [tex]$x^0 = 1 \quad\text{(option 3)}$[/tex]
- b. [tex]$\left(12x^3\right)^2 = 144x^6 \quad\text{(option 5)}$[/tex]
- c. [tex]$2x^{-2} = \frac{2}{x^2} \quad\text{(option 2)}$[/tex]
- d. [tex]$12x^2 \cdot (-5x^3) = -60x^5 \quad\text{(option 4)}$[/tex]
- e. [tex]$\frac{8x^{10}}{2x^2} = 4x^8 \quad\text{(option 1)}$[/tex]
1. For the expression
[tex]$$x^0,$$[/tex]
recall that any nonzero number raised to the zeroth power is 1. Thus,
[tex]$$x^0 = 1,$$[/tex]
which corresponds to option 3.
2. Next, consider
[tex]$$\left(12x^3\right)^2.$$[/tex]
Square the coefficient and the variable separately:
[tex]$$12^2 = 144,$$[/tex]
and
[tex]$$(x^3)^2 = x^{3 \cdot 2} = x^6.$$[/tex]
Therefore, the expression simplifies to
[tex]$$144x^6,$$[/tex]
which corresponds to option 5.
3. Now simplify
[tex]$$2x^{-2}.$$[/tex]
The negative exponent means we take the reciprocal of the base:
[tex]$$x^{-2} = \frac{1}{x^2}.$$[/tex]
So,
[tex]$$2x^{-2} = \frac{2}{x^2},$$[/tex]
which is option 2.
4. Consider the product
[tex]$$12x^2 \cdot (-5x^3).$$[/tex]
Multiply the coefficients:
[tex]$$12 \times (-5) = -60,$$[/tex]
and add the exponents on [tex]$x$[/tex]:
[tex]$$x^2 \cdot x^3 = x^{2+3} = x^5.$$[/tex]
Hence, the expression simplifies to
[tex]$$-60x^5,$$[/tex]
which corresponds to option 4.
5. Finally, simplify the quotient
[tex]$$\frac{8x^{10}}{2x^2}.$$[/tex]
Divide the coefficients:
[tex]$$\frac{8}{2} = 4,$$[/tex]
and subtract the exponents on [tex]$x$[/tex]:
[tex]$$x^{10} \div x^2 = x^{10-2} = x^8.$$[/tex]
Thus, the expression simplifies to
[tex]$$4x^8,$$[/tex]
which corresponds to option 1.
In summary, the simplified expressions and their matching options are:
- a. [tex]$x^0 = 1 \quad\text{(option 3)}$[/tex]
- b. [tex]$\left(12x^3\right)^2 = 144x^6 \quad\text{(option 5)}$[/tex]
- c. [tex]$2x^{-2} = \frac{2}{x^2} \quad\text{(option 2)}$[/tex]
- d. [tex]$12x^2 \cdot (-5x^3) = -60x^5 \quad\text{(option 4)}$[/tex]
- e. [tex]$\frac{8x^{10}}{2x^2} = 4x^8 \quad\text{(option 1)}$[/tex]
