Question

Evaluate the expression when \(x=2, y=3\), and \(z=5$$\sqrt[4]{16 x^{-4} y^{8} z^{4}}\)

Short answer

The value of the expression when x=2, y=3, and z=5 is 10.

Step-by-step solution

Substitute the values of x, y, and z in the expression

Substitute x=2, y=3, and z=5 into the equation \( \sqrt[4]{16 x^{-4} y^{8} z^{4}} \).This gives \( \sqrt[4]{16*2^{-4}*3^{8}*5^{4}} \).

Simplify the expression inside the root

Simplify the expression \(16*2^{-4}*3^{8}*5^{4}\) to \( \sqrt[4]{16*(1/2^{4})*(3^{8})*(5^{4}} \). When carrying out the exponent calculations, this further simplifies to \( \sqrt[4]{16*(1/16)*6561*625} \).

Calculate the multiplication inside the root

Calculate the multiplication operation \(16*(1/16)*6561*625\) gives \( \sqrt[4]{2560000} \).

Calculate the fourth root

Calculate the root \( \sqrt[4]{2560000} \) which equals 10.

Key concepts

Simplifying Expressions

When working with algebraic expressions, simplification is an essential step. Simplifying an expression involves reducing it to the simplest form without changing its value. This means combining like terms and utilizing algebraic properties to make the expression as straightforward as possible.
For example, in the given problem, we start with the expression \( \sqrt[4]{16 x^{-4} y^{8} z^{4}} \). Before fully solving it, it's crucial to simplify the expression inside the root by performing the division, multiplication, or cancellation wherever applicable.
One key aspect of simplification is recognizing how exponents and coefficients work. Simplifying ensures computations are efficient and less error-prone.

Substituting Values

Substituting values is a basic yet crucial process because it transforms a general expression into a specific number. It's like tailoring the math equation to fit a particular scenario. Given the variables (\(x\), \(y\), and \(z\)) and their respective values, \(x = 2\), \(y = 3\), and \(z = 5\), we substitute these directly into the expression.
This involves rewriting the expression \( \sqrt[4]{16 x^{-4} y^{8} z^{4}} \) to \( \sqrt[4]{16 \times 2^{-4} \times 3^{8} \times 5^{4}} \). Substitution simplifies the expression further because it allows us to work directly with numbers rather than variables.

Exponent Calculation

Calculating exponents is a fundamental skill in handling algebraic expressions. Exponents dictate how many times a number (the base) is multiplied by itself.
In our problem, exponent calculation comes into play when we rewrite \(2^{-4}\), \(3^{8}\), and \(5^{4}\). For example, \(2^{-4}\) is equivalent to \(\frac{1}{2^4}\), so we calculate \(2^4\) to get 16, leading to \(\frac{1}{16}\).
Similarly, \(3^8\) is found by multiplying 3 by itself seven more times, resulting in 6561. For \(5^4\), multiplying 5 four times results in 625. This efficient use of exponent rules is crucial for accurate and quick problem-solving.

Roots and Radicals

Roots and radicals are another vital component of evaluating expressions, especially when simplifying or evaluating expressions that include them. A root, such as a square root or a fourth root, "undoes" the operation of an exponent.
In this exercise, we encounter a fourth root, denoted by \( \sqrt[4]{} \). To evaluate \( \sqrt[4]{2560000} \), we need to determine what number raised to the fourth power gives 2560000. The fourth root here simplifies the expression, resulting in 10 because \(10^4 = 10000\).
Understanding how to work with radicals allows for better simplification and ultimately accurate results in solving complex algebraic expressions.