Certainly, let's go through each problem step-by-step to understand the solutions.
### 1] [tex]\( (-4)^2 \)[/tex]
When raising a negative number to an even power, the result is positive:
[tex]\[ (-4)^2 = 16 \][/tex]
### 2] [tex]\(-4^2\)[/tex]
This should be interpreted as [tex]\(-(4^2)\)[/tex]:
[tex]\[ -4^2 = -(4^2) = -(16) = -16 \][/tex]
### 3] [tex]\( 3^2 \cdot (-2)^3 \)[/tex]
First, calculate each component and then multiply:
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ (-2)^3 = -8 \][/tex]
[tex]\[ 9 \cdot (-8) = -72 \][/tex]
### 4] [tex]\( a^2 \cdot a^3 \)[/tex]
When multiplying exponential expressions with the same base, add the exponents:
[tex]\[ a^2 \cdot a^3 = a^{2+3} = a^5 \][/tex]
### 5] [tex]\( (d^4) (d^6) \)[/tex]
Similarly to Problem 4, add the exponents:
[tex]\[ d^4 \cdot d^6 = d^{4+6} = d^{10} \][/tex]
### 6] [tex]\( (x^4)^3 \)[/tex]
When raising a power to a power, multiply the exponents:
[tex]\[ (x^4)^3 = x^{4 \cdot 3} = x^{12} \][/tex]
### 7] [tex]\( (2x^2 y^3)^4 \)[/tex]
Distribute the exponent to each part inside the parentheses:
[tex]\[ (2x^2 y^3)^4 = 2^4 \cdot (x^2)^4 \cdot (y^3)^4 = 16 \cdot x^{8} \cdot y^{12} \][/tex]
[tex]\[ \text{Therefore, } (2x^2 y^3)^4 = 16 x^8 y^{12} \][/tex]
### 8] [tex]\( z^0 \cdot z^2 \)[/tex]
Any number to the power of 0 is 1, so first evaluate:
[tex]\[ z^0 = 1 \][/tex]
Then multiply:
[tex]\[ 1 \cdot z^2 = z^2 \][/tex]
### 9] [tex]\( x^{-3} \)[/tex]
A negative exponent means take the reciprocal of the base and apply the positive exponent:
[tex]\[ x^{-3} = \frac{1}{x^3} \][/tex]
### 10] [tex]\( 5^{-2} \)[/tex]
Similarly, take the reciprocal of 5 raised to the positive exponent:
[tex]\[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} = 0.04 \][/tex]
### 11] [tex]\( (-2x^2)(6x^3)(x^2) \)[/tex]
Multiply the coefficients and add the exponents for [tex]\(x\)[/tex]:
[tex]\[ (-2) \cdot 6 = -12 \][/tex]
[tex]\[ x^2 \cdot x^3 \cdot x^2 = x^{2+3+2} = x^7 \][/tex]
[tex]\[ \text{Therefore, } (-2x^2)(6x^3)(x^2) = -12x^7 \][/tex]
### 12] [tex]\(\frac{12a^{-2}}{4} \)[/tex]
First, divide the coefficients, then handle the exponent:
[tex]\[ \frac{12}{4} = 3 \][/tex]
[tex]\[ a^{-2} = \frac{1}{a^2} \][/tex]
So,
[tex]\[ \frac{12a^{-2}}{4} = 3 \cdot \frac{1}{a^2} = \frac{3}{a^2} \][/tex]
Each of these problems illustrates different rules of exponents that are essential for simplifying expressions and working with exponential functions.
