Question

Evaluate the expression. Write fractional answers in simplest form.\(\left(-3 \cdot 4^{2}\right)^{3}\)

Short answer

-110592

Step-by-step solution

Solve the operation inside the parentheses

According to the order of operations (PEMDAS/BODMAS), the operation inside the parentheses should be solved first. This includes finding the square of 4 and multiplying the resulting number by -3. That gives \(-3 \cdot 4^{2} = -3 \cdot 16 = -48\).

Solve the exponent

Now, the result of the operation inside the parentheses, which is -48, is raised to the power of 3. That gives \((-48)^{3} = -48 \cdot -48 \cdot -48 = -110592\).

Key concepts

Order of Operations

Understanding the order of operations is essential when evaluating expressions like \(\left(-3 \cdot 4^{2}\right)^{3}\), which require multiple steps to solve. This set of rules, often remembered through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right), or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), dictates the sequence in which calculations should be performed.

Initially, calculate operations inside parentheses or brackets, followed by dealing with exponents or 'orders.' Only after these should multiplication or division be handled, and finally, addition or subtraction. These rules are in place to avoid confusion and ensure that everyone gets the same result when solving mathematical expressions.

For example, in the expression given, the operation inside the parentheses is solved first (\(4^{2}\) is calculated to get 16), then this result is multiplied by -3. These steps are vital to maintaining mathematical consistency and arriving at the correct final result.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. The exponent signifies how many times the base is multiplied by itself. For instance, \(4^{2}\) indicates that 4 should be multiplied by itself exactly once, equating to 16.

In the exercise \(\left(-3 \cdot 4^{2}\right)^{3}\), after simplifying the expression inside the parentheses to -48, exponentiation is applied to this entire base. Here, \(\left(-48\right)^{3}\) requires -48 to be multiplied by itself two more times. This results in a much larger number, specifically \( -48 \cdot -48 \cdot -48 \), which equals to -110592.

Note that the negative sign plays a crucial role in exponentiation. An odd exponent (as in this case) will result in a negative outcome since multiplying an odd number of negative factors yields a negative product.

Simplifying Fractions

While the given exercise did not present a fraction, simplifying fractions is another key concept in algebra and arithmetic. Simplification involves reducing the fraction to its lowest terms, eliminating common factors from the numerator (the top number) and the denominator (the bottom number). To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).

For example, if you have the fraction \(\frac{8}{20}\), both 8 and 20 can be divided by their GCD of 4, resulting in the simplified fraction \(\frac{2}{5}\). Simplifying fractions aids in making calculations easier and results clearer. Though not used directly in exponentiation expressions, understanding how to simplify fractions can assist with the simplification process of expressions that result in fractional or decimal answers.