If `x= 10^(1.4), y = 10^(0.7) , and `x^(z) = y^(3)`, then what is the value of z?

Options

A. `0.5`
B. `0.66`
C. `1.5`
D. `2`

Answer

To solve the equation \( x^z = y^3 \) given \( x = 10^{1.4} \) and \( y = 10^{0.7} \), we can follow these steps:

### Step 1: Substitute the values of \( x \) and \( y \)
We know that:
\[ x = 10^{1.4} \quad \text{and} \quad y = 10^{0.7} \]
Substituting these into the equation \( x^z = y^3 \):
\[ (10^{1.4})^z = (10^{0.7})^3 \]

### Step 2: Apply the power of a power property
Using the property \( (a^m)^n = a^{m \cdot n} \), we can simplify both sides:
\[ 10^{1.4z} = 10^{0.7 \cdot 3} \]

### Step 3: Simplify the right side
Calculating the right side:
\[ 0.7 \cdot 3 = 2.1 \]
So we have:
\[ 10^{1.4z} = 10^{2.1} \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 1.4z = 2.1 \]

### Step 5: Solve for \( z \)
To find \( z \), divide both sides by \( 1.4 \):
\[ z = \frac{2.1}{1.4} \]

### Step 6: Simplify the fraction
Calculating \( \frac{2.1}{1.4} \):
\[ z = \frac{21}{14} = \frac{3}{2} = 1.5 \]

Thus, the value of \( z \) is:
\[ \boxed{1.5} \]

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