Question

Simplify the expression. $$ \left(4^{3 / 2} \cdot 4^{1 / 4}\right)^4 $$

Short answer

The simplified form of the given expression is 262144.

Step-by-step solution

Simplify the exponentiations inside parentheses

Begin with simplifying the base to the power of a fraction. The expression inside the parentheses is a product of two numbers, both are powers of 4. By recognizing 4^{3 / 2} is the square root of 4 cubed and 4^{1 / 4} is the fourth root of 4, we can evaluate them as \sqrt{4^3} = 8 and \sqrt[4]{4} = \sqrt{2}, respectively. So, the expression inside parentheses simplifies to 8 * \sqrt{2}.

Simplify the product inside the parentheses

Multiply the numbers derived in step 1 to simplify the expression inside the parentheses. Hence, 8 * \sqrt{2} = 16 \sqrt{2}.

Apply the exponent outside the parentheses

Next, multiply the power of 4 outside the parentheses to the number inside the parentheses 16 \sqrt{2}, resulting into (16 \sqrt{2})^4.

Simplify exponent of the whole number

16 to the power of 4 is 65536 and \(\sqrt{2}\) to the power of 4 is 4. So the final output is 65536 * 4.

Final Calculation

Finally, compute the multiplication 65536 * 4, which is 262144.

Key concepts

Rational Exponents

Rational exponents are a unique way of expressing roots and powers together in mathematical expressions. Instead of writing the square root of a number, we use a fraction as an exponent. For instance, when you see \(4^{3/2}\), this signifies that 4 is raised to a power that is both an exponent and a root. It is equal to the square root of 4 cubed, which can be broken down into steps:
\[4^{3/2} = \sqrt{4^3} = \sqrt{64} = 8\]
By using rational exponents, you can efficiently handle problems that involve both powers and roots, allowing you to make complex expressions simpler.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a power. The power indicates how many times the base is multiplied by itself. For example, in the expression \(4^{3/2}\), 4 is the base, and \(3/2\) is the exponent.
Key properties of exponentiation include:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Zero Exponent: Any non-zero base raised to the zero power is 1, e.g., \(a^0 = 1\)

When simplifying expressions, like in the original exercise, understanding how to manipulate and combine exponents is crucial.

Mathematical Roots

Mathematical roots are used to determine the original value that was raised to a certain power to reach a resulting number. Taking the square root of a number is equivalent to reversing the process of squaring that number. The concept of roots extends to cube roots, fourth roots, and beyond.
For instance, consider the expressions:

• The fourth root of 4, written as \(\sqrt[4]{4}\), simplifies to \(\sqrt{2}\).
• The cube of \(\sqrt{8}\) or the cube root of \(8\) returns the number 2.

Understanding mathematical roots helps break down expressions into simpler parts, essential for dealing with complex problems.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions to achieve a desired form or to find solutions. This procedure often includes techniques such as combining like terms, factoring, and expanding.
In the context of the original exercise, algebraic manipulation was used after simplifying individual components. After breaking down the expression \(\left(4^{3/2} \cdot 4^{1/4}\right)^4\) to \(16 \sqrt{2}\), further manipulation involved applying powers to results:

• Raising 16 to the power of 4 gives 65536.
• Raising \(\sqrt{2}\) to the power of 4 converts to 4.

This kind of manipulation is essential for rewriting expressions in a simpler or more insightful form, making problems easier to solve.