Problem 67 - Contemporary Precalculus

Question

Simplify the expression without using a calculator. Your answer should not have any radicals in it. \(\sqrt{u^{4}}(u \text { any real number })\) 68

Step-by-Step Solution

Answer

Question: Simplify the expression \(\sqrt{u^{4}}\) without using a calculator. Answer: \(u^2\)

Step 1: Identify the property

In this case, we have an expression in the form of \(\sqrt{u^4}\). We need to use the property of square roots that states \(\sqrt{a^2} = a\), where a is a real number.

Step 2: Apply the property

Now, let's apply this property on the given expression. We have: \(\sqrt{u^{4}}\). Notice that \(u^{4}\) is a perfect square (\((u^2)^2\)), so we can apply the property: \(\sqrt{u^{4}} = u^{4/2} = u^2\)

Step 3: Write the simplified expression

So, the simplified expression without any radicals is: \(u^2\)

Key Concepts

Understanding Square Roots

The concept of square roots is fundamental in algebra. When we talk about a square root, we're referring to a number that, when multiplied by itself, equals the original number. For instance, the square root of 9 is 3 because

• 3 multiplied by 3 equals 9.
• A square root is generally depicted by the symbol \( \sqrt{} \).

This can be understood as asking, "What number gives us this product when squared?". In expressions, the square root serves to simplify and elucidate the nature of numbers or variables involved. It's important to remember that the square root

• always refers to the positive root by default.

For example, for a variable expression like \( \sqrt{u^4} \), we seek to find a value that, when squared, returns \( u^4 \). If \( a^2 = a \), then the most direct simplification of \( \sqrt{a^2} \) is \( a \), assuming \( a \) is non-negative. This principle helps simplify expressions involving radicals.

Exponents and Their Properties

Exponents are a powerful tool in algebra. They represent repeated multiplication of a number by itself. For instance, with \( u^4 \), the number \( u \) is multiplied by itself four times, written as \( u \times u \times u \times u \). Understanding exponents is essential for algebraic simplification because they help reduce the complexity of expressions. The key properties of exponents you'll encounter include:

• Product of powers: Combining like bases with their exponents, \( a^m \times a^n = a^{m+n} \).
• Power of a power: Simplifying when raising an exponent to another power, \( (a^m)^n = a^{mn} \).
• Division of exponentials: Simplifies when dividing like bases, \( a^m / a^n = a^{m-n} \).

In the exercise, identifying \( u^4 \) as a perfect square plays into understanding the property of powers. We know \( (u^2)^2 = u^4 \), allowing us to simplify the radical expression to \( u^2 \) by applying the simplification property of exponents where \( n = 4 \) here.

Perfect Squares and Their Role in Simplification

The term "perfect square" refers to the square of an integer or a variable expression, signifying a number that can be expressed as another number squared. Understanding perfect squares is crucial. They allow for simplification of square-root expressions, as seen when the original expression is already a perfect square, often bypassing complex calculations.For example, numbers like 1, 4, 9, 16—produced by squaring 1, 2, 3, 4—are perfect squares. In algebra, \( u^4 \) is a perfect square because it can be expressed as the square of \( u^2 \). Recognizing \( u^4 \) as a perfect square tells us:

• It can be simplified without a radical by acknowledging the expression as \( (u^2)^2 \).
• Removing the root results in a more direct form: \( u^2 \).

Identifying and utilizing perfect squares streamlines problem-solving, turning what could seem like a complex problem into a clear-cut solution by reducing the square root to its base form.