Use the properties of exponents to simplify the expression: \( \left(a^{6}\right)^{2} a^{7} \) a.) \( a^{19} \) b.) \( a^{15} \) c.) \( a^{84} \) d.) \( a^{56} \)

First, apply the power of a power property of exponents, which states that \((x^m)^n = x^{m \cdot n}\). This means we can simplify \((a^6)^2\) to \(a^{12}\). Now combine it with \(a^7\) using the property \(x^m \cdot x^n = x^{m+n}\): \[a^{12} \cdot a^{7} = a^{12+7} = a^{19}.\] So the correct answer is \( \text{a.) } a^{19} \). Understanding the properties of exponents can lead to faster calculations in algebra. It's like having a secret shortcut! Properties such as \(x^0 = 1\) (for \(x \neq 0\)) and \(x^{-n} = \frac{1}{x^n}\) are essential tools that transform complicated expressions into manageable ones, making math less daunting. If you want to solidify your understanding of exponents, consider checking out online math platforms like Khan Academy or educational YouTube channels. They often provide engaging videos that break down complex concepts into bite-sized pieces, with plenty of examples to practice on your own. Happy learning!