Questions: Provide values of s and t, that will make each equation true. a. x^2 * x^t = x^10 b. y^s = 1 c. (a^t)^s = a^12 d. (k^5 y^t)^2 = k^4 y

Transcript text: Provide values of $s$ and $t$, that will make each equation true. a. $x^{2} \cdot x^{t}=x^{10}$ b. $y^{s}=1$ c. $\left(a^{t}\right)^{s}=a^{12}$ d. $\left(k^{5} y^{t}\right)^{2}=k^{4} y$

Solution

Solution Steps

To solve these equations, we need to apply the properties of exponents.

a. For the equation $x^2 \cdot x^t = x^{10}$, use the property that when multiplying like bases, you add the exponents.

b. For the equation $y^s = 1$, recall that any number raised to the power of 0 is 1.

c. For the equation $(a^t)^s = a^{12}$, use the property that when raising a power to another power, you multiply the exponents.

Solution Approach

a. Set up the equation $2 + t = 10$ and solve for $t$.
b. Set up the equation $s = 0$ since any number to the power of 0 is 1.
c. Set up the equation $t \cdot s = 12$ and solve for $t$ and $s$.

Step 1: Solve for $ t $ in Equation (a)

From the equation $ x^2 \cdot x^t = x^{10} $, we set up the equation: \[ 2 + t = 10 \] Solving for $ t $, we find: \[ t = 10 - 2 = 8 \]

Step 2: Solve for $ s $ in Equation (b)

From the equation $ y^s = 1 $, we know that: \[ s = 0 \] since any number raised to the power of 0 equals 1.

Step 3: Solve for $ t $ and $ s $ in Equation (c)

From the equation $ (a^t)^s = a^{12} $, we set up the equation: \[ t \cdot s = 12 \] Substituting $ s = 0 $ into the equation, we see that $ t $ can be any value since $ 0 \cdot t = 0 $ does not equal 12. Therefore, we express $ t $ in terms of $ s $: \[ t = \frac{12}{s} \] This indicates that $ t $ is dependent on the value of $ s $.