Problem 110 - College Algebra

Question

Evaluate each expression. $$\sqrt[3]{\sqrt{\sqrt{169}+\sqrt{9}}+\sqrt{\sqrt[3]{1000}+\sqrt[3]{216}}}$$

Step-by-Step Solution

Answer

The result of evaluating the expression is 2.

Step 1: Calculate the innermost square roots

Calculate the square roots inside the main cube root. For \(\sqrt{169}\), the result is 13. For \(\sqrt{9}\), the result is 3.

Step 2: Sum the results and calculate the outer square root

Sum the results from Step 1 and compute the square root: \(\sqrt{13 + 3} = \sqrt{16} = 4\)

Step 3: Calculate the innermost cube roots

Compute the cube roots inside the main square root. For \(\sqrt[3]{1000}\), the result is 10. For \(\sqrt[3]{216}\), the result is 6.

Step 4: Sum the results and calculate the outer square root

Sum the results from Step 3 and compute the square root: \(\sqrt{10 + 6} = \sqrt{16} = 4\)

Step 5: Sum the results from the square root operations and extract the cube root

Add the results from Step 2 and Step 4 and compute the cube root: \(\sqrt[3]{4 + 4} = \sqrt[3]{8} = 2\)

Key Concepts

Cube Roots

Cube roots are the numbers that, when multiplied by themselves three times, give the original number. For instance, the cube root of 8 is 2 since multiplying 2 three times (i.e., 2 \( \times \) 2 \( \times \) 2) results in 8.

• To express this, use the notation \( \sqrt[3]{8} = 2 \).
• This symbol, \(\sqrt[3]{}\), indicates the cube root.

Finding cube roots often involves breaking down the number into its prime factors. For example, 1000 can be broken down into \(2^3 \times 5^3\), making \(10 \times 10 \times 10 = 1000\). Thus, \(\sqrt[3]{1000} = 10\).
Cube roots commonly appear in various mathematical problems, especially those involving geometry or when interpreting the volume of three-dimensional shapes. Understanding them is essential for solving expressions where numbers are raised to the third power.

Square Roots

Square roots are fundamental in mathematics, representing what number can be squared to produce the initial value. For example, the square root of 169 is 13 because \( 13 \times 13 = 169 \).

• Regularly, the square root is depicted using the symbol \(\sqrt{}\).
• Finding square roots involves recognizing perfect squares such as 9, 16, 25, which are easily found because their square roots are whole numbers like 3, 4, and 5 respectively.

A square root is often needed when dealing with problems in areas such as algebra and geometry.
It helps simplify expressions or find unknown sides in right triangles using the Pythagorean theorem. Knowing how to handle square roots simplifies delving into more complicated problems, like those involving nested radicals.

Nested Radicals

Nested radicals refer to radical expressions within other radical expressions. They can be tricky to simplify but follow a systematic approach.

• The simplest form of expressing nested radicals is solving them layer by layer, beginning from the innermost radical.
• In expression threads like \( \sqrt[3]{\sqrt{\sqrt{169} + \sqrt{9}} + \sqrt{\sqrt[3]{1000} + \sqrt[3]{216}}} \), it’s crucial to tackle each radical systematically starting from the inside.

These expressions often involve handling both square and cube roots together, adding complexity to the process. To efficiently solve nested radicals:- First, address each inner square root or cube root as individual units.- Calculate the sum of these units.- Proceed to evaluate any outer radicals.
Understanding nested radicals is particularly valuable in higher-level mathematics, where complex calculations intersect in algebraic manipulations and simplifications.