Problem 69 - Algebra 2

Question

Which expression does NOT simplify to \(-10 ?\) $$ \begin{array}{ll}{\text { A. }-\sqrt[3]{1000}} & {\text { B. } \sqrt{25} \cdot \sqrt[3]{-8}} \\ {\text { C. }-\sqrt{25} \cdot \sqrt[5]{-32}} & {\text { D. } \sqrt[3]{-125} \cdot \sqrt[4]{16}}\end{array} $$

Step-by-Step Solution

Answer

Option C does not simplify to -10.

Step 1: Simplify Option A

First, simplify expression A, which is \( -\sqrt[3]{1000} \). The cube root of 1000 is 10. Therefore the expression becomes -10.

Step 2: Simplify Option B

The second expression is \( \sqrt{25} * \sqrt[3]{-8} \). First separate positive root and negative root. \(\sqrt{25}\) is 5 and the third root of -8 is -2. Therefore the expression simplifies to \(5*-2\), which equals -10.

Step 3: Simplify Option C

Simplify the expression of option C: \( -\sqrt{25}*\sqrt[5]{-32} \). Splitting this, sqare root of 25 is 5 & fifth root of -32 is -2 , then the expression becomes \(-5*-2 = 10\). Contrary to our expected answer of -10. Option C does not simplify to -10.

Step 4: Simplify Option D

Finally, simplify the fourth expression, which is \( \sqrt[3]{-125} * \sqrt[4]{16} \). Here the cube root of -125 is -5 and the fourth root of 16 is 2. Therefore the resulting expression is -5*2 = -10.

Key Concepts

Cube Root

Understanding the cube root of a number is essential for simplifying expressions. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 1000 is 10, because \(10 \times 10 \times 10 = 1000\).

• Cube roots can be calculated for both positive and negative numbers.
• For negative numbers, the cube root will also be negative. For instance, the cube root of -8 is -2, since \((-2) \times (-2) \times (-2) = -8\).
• This is unlike square roots, where negative numbers under the root would result in complex numbers.

Being comfortable with cube roots allows you to manage expressions like \(-\sqrt[3]{1000}\), which simplifies directly to -10 after finding the cube root of 1000, demonstrating its application in problems like ours.

Square Root

A square root simplifies numbers by finding a value that, when multiplied by itself, returns to the original number. The square root of 25, for example, is 5 because \(5 \times 5 = 25\).

• Square roots generally deal with non-negative numbers as the principal value.
• They can simplify expressions by reducing terms into easier numbers, making calculations easier.
• In equations, knowing the square root helps identify factors and simplifies more complex mathematical expressions.

For expression B, recognizing \(\sqrt{25}\) allows simplifying it to 5, which is essential for further calculating the result when combined with other roots, like the cube root of -8.

Negative Numbers

Negative numbers have unique properties that affect how expressions are simplified.They influence signs within expressions and can reverse expected outcomes like in multiplication. Negative numbers when multiplied by a positive number result in a negative product, and vice versa for two negatives, which result in a positive product.

• This rule is essential in problems where negative signs change the final evaluated answer.
• For example, in the expression \( -\sqrt{25} \cdot \sqrt[5]{-32} \), the negative before the square root switches the outcome of the expression from -10 to 10.

Understanding this property allows one to correctly simplify complex expressions involving both roots and negative signs.

Fifth Root

The fifth root of a number finds a value that results in the original number when raised to the power of five. For example, the fifth root of -32 is -2, since \((-2)^5 = -32\).

• Fifth roots are less commonly used but important for high-level simplifications.
• They can be significant in simplifying expressions in algebra and calculus.
• For negative numbers, the result remains negative, unlike in square roots.

In expression C, knowing \(\sqrt[5]{-32} = -2\) helps identify how terms interact within a formula when combined with others, such as \(-\sqrt{25}\), illuminating the expression's true simplified value.