Problem 14 - Elementary Algebra

Question

For Problems \(1-20\), find the value of each numerical expression. For example, \(2^{4}=2 \cdot 2 \cdot 2 \cdot 2=16\). $$ -\left(\frac{3}{2}\right)^{3} $$

Step-by-Step Solution

Answer

The value is \(-\frac{27}{8}\).

Step 1: Understand the Base and Exponent

The expression \(-\left(\frac{3}{2}\right)^{3}\) involves raising the base, which is \(\frac{3}{2}\), to the power of \(3\). The negative sign outside the parentheses indicates that the whole expression's value should be multiplied by \(-1\).

Step 2: Calculate the Exponent

Calculate \(\left(\frac{3}{2}\right)^{3}\). This means multiplying the fraction \(\frac{3}{2}\) by itself three times:\[\left(\frac{3}{2}\right) \times \left(\frac{3}{2}\right) \times \left(\frac{3}{2}\right)\]First, multiply the numerators:\(3 \times 3 \times 3 = 27\).Then, multiply the denominators:\(2 \times 2 \times 2 = 8\).Thus, \(\left(\frac{3}{2}\right)^{3} = \frac{27}{8}\).

Step 3: Apply the Negative Sign

Now apply the negative sign to \(\frac{27}{8}\), as indicated by the original expression. Multiply the fraction by \(-1\):\[-\left(\frac{27}{8}\right) = \frac{-27}{8}\].This yields the final value of the expression.

Key Concepts

Fraction Multiplication

When multiplying fractions, always remember that you are dealing with both the numerators and the denominators separately. The process is straightforward:

• Multiply the numerators of each fraction together. This gives you the numerator of your answer.
• Multiply the denominators of each fraction involved. This will be the denominator of your final result.

For example, if you are multiplying \( \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} \), first find the product of the numerators:- Multiply \(3 \times 3 \times 3 = 27\).Next, calculate the product of the denominators:- Multiply \(2 \times 2 \times 2 = 8\).Thus, \((\frac{3}{2})^3 = \frac{27}{8}\), showcasing how fraction multiplication works in practice. Breaking fractions into their parts simplifies even the most complex-looking problems. This method is essential when dealing with exponents applied to fractions.

Negative Sign

Dealing with negative signs in numerical expressions is crucial for getting the correct result. A negative sign outside a mathematical operation can be thought of as multiplying by \(-1\). Here's how it impacts calculations:

• If an expression is negative outside the parentheses, as in \(-\left(\frac{3}{2}\right)^{3}\), it tells us to take the result of the expression and change its sign.
• This can directly influence the final result. Apply the negative sign after fully computing the mathematical expression inside the parentheses.

So after computing \(\left(\frac{3}{2}\right)^3\) and finding \(\frac{27}{8}\), you apply the negative sign:- Multiply \(\frac{27}{8}\) by \(-1\), resulting in \(-\frac{27}{8}\).Remember, the negative outside affects the whole expression, so ensure that it is applied last to avoid mistakes.

Numerical Expressions

Understanding numerical expressions involves knowing how different mathematical operations within them affect the outcome. These expressions can include:

• Exponents, which tell you how many times to multiply a number by itself.
• Negative signs, which alter the value of expressions depending on their position.

The key is to follow the operations systematically, respecting the order of operations:- Parentheses- Exponents- Multiplication/Division, left to right- Addition/Subtraction, left to right
In the example \(-\left(\frac{3}{2}\right)^3\), begin by computing the exponent:- Multiply the fraction by itself as instructed by the exponent.
Next, address operations outside the parentheses like negative signs, applying them after performing other calculations inside. By dissecting each part of the expression and handling them step-by-step, you'll manage even complex calculations with ease. Remember always to approach such expressions methodically.