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Chapter 1: Problem 74

Evaluate. $$ -30 \div t(t+4)^{2}, \text { for } t=-6 $$

Short Answer

Expert verified
The value is \(\frac{5}{4}\).

Step by step solution

01

Substitute the value of t

Replace the variable \(t\) with the given value \(-6\) in the expression. The expression becomes:e\(-30 \div (-6)(-6+4)^{2}\)
02

Simplify inside the parentheses

First, calculate the value inside the parentheses. Thus,e\(-6+4 = -2\)
03

Evaluate the exponent

Next, evaluate the square of \(-2\). Therefore,e\((-2)^{2} = 4\)
04

Perform the multiplication

Now multiply \(-6\) by the result of the squared term.e\(-6 \cdot 4 = -24\)
05

Perform the division

Finally, divide \(-30\) by \(-24\). So,e\(-30 \div -24 = \frac{5}{4}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution
In this exercise, 'substitution' refers to replacing a variable with a given number. Here, the variable is \( t \). You need to replace \( t \) with \( -6 \). So the expression \( -30 \div t(t+4)^{2} \) turns into \( -30 \div (-6)(-6+4)^{2} \).

This step is crucial because it turns the algebraic expression into a numerical one you can simplify.
Simplifying Expressions
Simplifying expressions means reducing them to their simplest form. Start by handling the brackets, as operations inside them take precedence. In the expression, you have \( (-6+4) \). Calculate this first:

\[ -6 + 4 = -2 \]

Now substitute \( -2 \) back into the expression, making it \( -30 \div (-6) (-2)^{2} \).
Exponents
Exponents are a way of indicating repeated multiplication. Here, you have \( (-2)^{2} \), which means multiplying \( -2 \) by itself. So:

\[ (-2) \times (-2) = 4 \]

This is because multiplying two negative numbers results in a positive number.
Division
Division means splitting a number into equal parts. To simplify \( -30 \div (-6) \times 4 \), follow these steps:

  • Multiply the non-division part first: \( -6 \times 4 = -24 \).
  • Now, handle the division: \( -30 \div -24 \).

When dividing negative numbers, the negatives cancel out, making the result positive. So:

\[ -30 \div -24 = \frac{5}{4} \]

And that's your final answer!

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