Work out the values of $$ a $$ and $$ b $$ in the equality $$ \left(2 x^{3}\right)^{a}=b x^{12} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\left(2 x^{3}\right)^{a} = b x^{12}
$$

we need to determine the values of $$ a $$ and $$ b $$ such that the equation holds true for all values of $$ x $$.

**Step 1: Expand the Left Side**

First, expand the left side of the equation:

$$
\left(2 x^{3}\right)^{a} = 2^{a} \cdot x^{3a}
$$

**Step 2: Compare Exponents of $$ x $$**

The right side of the equation is $$ b x^{12} $$. For the equation to hold for all $$ x $$, the exponents of $$ x $$ on both sides must be equal:

$$
3a = 12
$$

Solving for $$ a $$:

$$
a = \frac{12}{3} = 4
$$

**Step 3: Compare the Coefficients**

Now, equate the coefficients of the expressions:

$$
2^{a} = b
$$

Substitute the value of $$ a $$ we found:

$$
2^{4} = b \implies b = 16
$$

**Conclusion**

The values that satisfy the equation are:

$$
a = 4 \quad \text{and} \quad b = 16
$$

**Final Answer:**

After simplifying, the solution is a = 4 and b = 16.
$$ a = 4 $$ and $$ b = 16 $$.

Work out the values of \( a \) and \( b \) in the equality \[ \left(2 x^{3}\right)^{a}=b x^{12} \]