The equation (z^(3+t))^(4)=z^(20) is true for all values of z. What is the value of t ?

Apply Property of Exponents: We are given the equation (z^(3+t))^(4) = z^(20). We need to find the value of t that makes this equation true for all values of z. To solve for t, we will use the property of exponents that states (a^b)^c = a^(b*c). Apply this property to the left side of the equation. (z^(3+t))^4 = z^((3+t)*4)Equating Exponents: Now we have z^((3+t)*4) = z^(20). Since the bases are the same and the equation holds for all values of z, we can equate the exponents. (3+t)*4 = 20Solving for t: Next, we solve for t. 3*4 + t*4 = 20 12 + 4t = 20Isolating t Term: Subtract 12 from both sides of the equation to isolate the term with t. 4t = 20 - 12 4t = 8Final Solution for t: Finally, divide both sides by 4 to solve for t. t = 8 / 4 t = 2

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Q. The equation

\left(z^{3+t}\right)^{4}=z^{20}

is true for all values of

z

. What is the value of

t

Apply Property of Exponents: We are given the equation $(z^{(3+t)})^{4} = z^{20}$ . We need to find the value of $t$ that makes this equation true for all values of $z$ . To solve for $t$ , we will use the property of exponents that states $(a^{b})^{c} = a^{(b*c)}$ . Apply this property to the left side of the equation. $(z^{(3+t)})^{4} = z^{((3+t)*4)}$
Equating Exponents: Now we have $z^{(3+t)\cdot 4} = z^{20}$ . $\newline$ Since the bases are the same and the equation holds for all values of $z$ , we can equate the exponents. $\newline$ $(3+t)\cdot 4 = 20$
Solving for $t$ : Next, we solve for $t$ . $3 \times 4 + t \times 4 = 20$ $12 + 4t = 20$
Isolating t Term: Subtract $12$ from both sides of the equation to isolate the term with $t$ . $\newline$ $4t = 20 - 12$ $\newline$ $4t = 8$
Final Solution for t: Finally, divide both sides by $4$ to solve for t. $\newline$ t = $\frac{8}{4}$ $\newline$ t = $2$