Problem 129 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. There are many exponential expressions that are equal to \(36 x^{12},\) such as \(\left(6 x^{6}\right)^{2},\left(6 x^{3}\right)\left(6 x^{9}\right), 36\left(x^{3}\right)^{9},\) and \(6^{2}\left(x^{2}\right)^{6}\)

Step-by-Step Solution

Answer

The expressions \(\left(6 x^{6}\right)^{2}\), \(\left(6 x^{3}\right)\left(6 x^{9}\right)\), and \(6^{2}\left(x^{2}\right)^{6}\) are equivalent to \(36x^{12}\). The expression \(36\left(x^{3}\right)^{9}\) is not equal to \(36x^{12}\).

Step 1: Compare first expression \(\left(6 x^{6}\right)^{2}\) with \(36x^{12}\)

Apply the power of a power law: When raising a power to a power, multiply the exponents. Here, [6*(x^6)]^2 equals 6^2 * (x^6)^2, which simplifies to 36 * x^12 which matches the original expression.

Step 2: Compare second expression \(\left(6 x^{3}\right)\left(6 x^{9}\right)\) with \(36 x^{12}\)

For multiplication, you add the exponents if the bases are the same. So, \(6x^3 * 6x^9 \) becomes \( 36x^{12}\) which matches the original expression.

Step 3: Compare third expression \(36\left(x^{3}\right)^{9}\) with \(36 x^{12}\)

Using the power of a power rule, \((x^3)^9\) becomes \(x^{27}\). So, \(36x^{27}\) does not equal \(36x^{12}\). Therefore, this expression does not match the original expression.

Step 4: Compare fourth expression \(6^{2}\left(x^{2}\right)^{6}\) with \(36 x^{12}\)

Once again, applying the power of a power rule, \((x^2)^6\) becomes \(x^{12}\). However, \(6^2x^{12}\) becomes \(36x^{12}\). Therefore, this expression matches the original expression.