Problem 78 - Elementary Algebra

Question

For the following problems, simplify each expression by removing the radical sign. $$ \sqrt{169 w^{4} z^{6}(m-1)^{2}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \(13w^2z^3(m-1)\).

Step 1: Find the square root of the constant

The first step is to find the square root of the constant 169. The square root of 169 is 13.

Step 2: Simplify the variables with even exponents

Next, we need to simplify the variables with even exponents inside the square root. For each variable, we will divide the exponent by 2. The square root of \(w^{4}\) is \(w^{4/2}=w^2\). The square root of \(z^{6}\) is \(z^{6/2}=z^3\). Since the exponent of \((m-1)^{2}\) is already 2 (which is even), we can find its square root by simply removing the exponent: \((m-1)^{2/2} = (m-1)\).

Step 3: Combine all simplified parts

Finally, we combine all the simplified parts of the expression: $$ \sqrt{169 w^{4} z^{6}(m-1)^{2}} = 13w^2z^3(m-1) $$ So, the simplified expression without the radical sign is \(13w^2z^3(m-1)\).

Key Concepts

Square Roots

The square root is a fundamental mathematical concept that involves finding a number which, when multiplied by itself, gives the original number. It's indicated by the radical symbol \( \sqrt{} \). For example, the square root of 169 is 13, because \(13 \times 13 = 169\).
The goal of square roots is often to simplify expressions, especially when they involve constants and variables raised to powers. For instance, a square root can remove an exponent if it's perfectly squared, simplifying computations and algebraic manipulations.
Square roots are widely used in various calculations, including geometry, physics, and algebra. Being comfortable with square roots can simplify working with quadratic equations and complex numbers.

Exponents

Exponents play a crucial role in mathematics, providing a way to express repeated multiplication of the same number. In the expression \(a^n\), \(a\) is the base and \(n\) is the exponent, meaning \(a\) is multiplied by itself \(n\) times.
Simplifying expressions with exponents involves understanding certain rules. One important principle is that the square root of a number or a variable can halve its exponent. This is what happened in the original problem, where expressions like \(w^4\) and \(z^6\) are simplified to \(w^2\) and \(z^3\) respectively by taking their square roots.
Key rules also include:

• Multiplying like bases: add exponents \(a^m \times a^n = a^{m+n}\).
• Dividing like bases: subtract exponents \(\frac{a^m}{a^n} = a^{m-n}\).
• Taking a power of a power: multiply exponents \((a^m)^n = a^{m \times n}\).

Understanding these rules will enable you to manipulate algebraic expressions more effectively.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value. They are foundational in mathematics for forming equations and expressing general relationships.
When simplifying algebraic expressions, particularly those involving square roots or exponents, it's essential to follow systematic steps. Identify constants that can be simplified, such as finding perfect squares for square roots. Simplify variables with even exponents by dividing the exponents by two when dealing with square roots, as seen in the original problem.
Breaking down complex expressions into their simpler parts can make algebra tasks more manageable. Doing so not only clarifies the expression itself but also helps when performing further operations, like solving equations or graphing.
Mastery of simplifying algebraic expressions is crucial for success in higher-level math courses, where these skills are applied to more advanced topics, such as calculus and linear algebra.