Question

Write each expression as a single power of 4. a) \((\sqrt{16})^{2}\) b) \(\sqrt[3]{16}\) c) \(\sqrt{16}(\sqrt[3]{64})^{2}\) d) \((\sqrt{2})^{8}(\sqrt[4]{4})^{4}\)

Short answer

a) 4, b) 4^{2/3}, c) 4^{3}, d) 4^{3}

Step-by-step solution

Simplify \(\sqrt{16}\)

The square root of 16 is 4, because \(4^2 = 16\). So \(\sqrt{16} = 4\).

Simplify \(\sqrt[3]{16}\)

The cube root of 16 can be expressed as \(16^{1/3}\). \(16 = 4^2\), so \(\sqrt[3]{16} = (4^2)^{1/3} = 4^{2/3}\).

Simplify the first part of the product in (c)

From Step 1, \(\sqrt{16} = 4\).

Simplify the second part of the product in (c)

From Step 2, \(\sqrt[3]{64} = 4\), so \( (\sqrt[3]{64})^2 = 4^2 = 16\).

Combine the terms in (c)

Multiply the two parts: \(4 \times 16 = 4^1 \times 4^2 = 4^{3}\).

Simplify the first product in (d)

\(\sqrt{2} = 2^{1/2}\). Thus, \( (\sqrt{2})^8 = (2^{1/2})^8 = 2^4 = 16\).

Simplify the second product in (d)

\(\sqrt[4]{4} = 4^{1/4}\). Thus, \( (\sqrt[4]{4})^4 = (4^{1/4})^4 = 4\).

Combine the terms in (d)

Multiply the two parts: \(16 \times 4 = 4^2 \times 4^1 = 4^{3}\).

Key concepts

Square Roots

Square roots are fundamental in mathematics. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4, because 4 times 4 equals 16. When written in mathematical notation, it looks like \(\sqrt{16} = 4\). To simplify expressions involving square roots, identify the factor that, when squared, returns the number under the root.

For example:
1. \(\sqrt{25} = 5\) because \(5 \times 5 = 25\)
2. \(\sqrt{81} = 9\) because \(9 \times 9 = 81\)

Square roots can also be used to simplify more complex expressions by isolating the square root and solving it first.

Cube Roots

Cube roots are similar to square roots but involve finding a value that, when cubed, gives the original number. For instance, the cube root of 64 is 4 because \(4^3 = 64\). Cube roots are denoted using the radical symbol with an index of three: \(\sqrt[3]{64} = 4\). Simplifying cube roots can be a bit more intricate than square roots. It often involves expressing the number as a power and then applying the root.

For example:
1. Let's simplify \(\sqrt[3]{8}\)
- 8 can be expressed as \(2^3\)
- Therefore, \(\sqrt[3]{8} = \sqrt[3]{2^3} = 2\)
2. Simplify \(\sqrt[3]{27}\)
- 27 can be expressed as \(3^3\)
- Hence, \(\sqrt[3]{27} = \sqrt[3]{3^3} = 3\).

This technique is especially useful when simplifying expressions involving cube roots and higher-order roots.

Exponential Expressions

Exponential expressions involve numbers raised to a power. The base is the number being multiplied, and the exponent tells how many times to multiply the base by itself. For example, in the expression \(4^3\), 4 is the base and 3 is the exponent, which means \(4 \times 4 \times 4 = 64\).

Exponents can be manipulated using different rules:
1. \(a^m \times a^n = a^{m+n}\) - When multiplying like bases, add the exponents.
2. \(\frac{a^m}{a^n} = a^{m-n}\) - When dividing like bases, subtract the exponents.
3. \( (a^m)^n = a^{m \times n}\) - When raising a power to another power, multiply the exponents.
4. \(a^{-n} = \frac{1}{a^n}\) - Negative exponents indicate the reciprocal of the base raised to the positive exponent.

In the provided exercise, these rules are frequently used to combine and simplify terms.

Simplification Steps

Simplification is an essential process in mathematics that makes expressions easier to work with. Here are the basic steps to simplify exponential expressions:
1. **Identify the bases**: Find the numbers or variables being raised to a power.
2. **Apply exponent rules**: Use the rules of exponents (addition, subtraction, multiplication) to combine and reduce the terms.
3. **Simplify roots**: Convert square and cube roots into their exponential form to facilitate further simplification.
4. **Combine like terms**: Once everything is expressed in the simplest exponential form, combine like terms.

In our exercise:
- For expression (a), simplify \(\sqrt{16}\) first, then square the result.
- For expression (b), express the cube root in exponential form and simplify.
- For expression (c), simplify both the square root and the cube root part first. Multiply them together last.
- For expression (d), express the square root and fourth root parts in exponential form. Simplify and combine all terms in the end.

This step-by-step approach ensures correct and clear simplification. By following these methods, complex expressions can be systematically broken down for easier handling.