Problem 45
Question
Simplify. $$ \sqrt{169 w^{8} y^{10}} $$
Step-by-Step Solution
Verified Answer
13 \(w^4 y^5\)
1Step 1: Identify the components inside the square root
Inside the square root, we have the expression: 169, \(w^{8}\), and \(y^{10}\).
2Step 2: Simplify the square root of the constant
The square root of 169 is 13, because \(13^2 = 169\).
3Step 3: Simplify the square root of \(w^{8}\)
Applying the power rule for square roots, we have \(\sqrt{w^8} = w^{8/2} = w^4\).
4Step 4: Simplify the square root of \(y^{10}\)
Similarly, applying the power rule for square roots, we have \(\sqrt{y^{10}} = y^{10/2} = y^5\).
5Step 5: Combine the simplified components
Combining the simplified parts, we get the final expression \(13 w^4 y^5\).
Key Concepts
square rootpower rulealgebra
square root
A square root is a mathematical operation used to find a number that, when multiplied by itself, gives the original number. For instance, the square root of 169 is 13 because 13, when multiplied by itself (13 × 13), equals 169. The square root symbol is denoted as \(\sqrt{...}\).
Square Root of a Constant
In our specific example, \(\sqrt{169}\), we recognize that 13 is the number which, when squared (13^2), yields 169. So, \(\sqrt{169} = 13\).
Square Root of Variables with Exponents
The square root operation extends to algebraic terms as well. Such as simplifying \(\sqrt{w^{8}}\) and \(\sqrt{y^{10}}\). This involves using the power rule, which we'll unpack in the next section.
Square Root of a Constant
In our specific example, \(\sqrt{169}\), we recognize that 13 is the number which, when squared (13^2), yields 169. So, \(\sqrt{169} = 13\).
Square Root of Variables with Exponents
The square root operation extends to algebraic terms as well. Such as simplifying \(\sqrt{w^{8}}\) and \(\sqrt{y^{10}}\). This involves using the power rule, which we'll unpack in the next section.
power rule
The power rule for square roots is an essential tool in algebra. It helps simplify square roots that involve exponents. The rule states that \(\sqrt{a^n} = a^{n/2}\). This means you take the exponent (n) and divide it by 2.
Applying the Power Rule
In our example, for \(\sqrt{w^{8}}\), applying the power rule gives \(w^{8/2} = w^4\). Similarly, for \(\sqrt{y^{10}}\), we have \(y^{10/2} = y^5\).
By utilizing this rule, we can simplify complex square roots containing variables, making them much easier to work with in algebraic expressions.
Applying the Power Rule
In our example, for \(\sqrt{w^{8}}\), applying the power rule gives \(w^{8/2} = w^4\). Similarly, for \(\sqrt{y^{10}}\), we have \(y^{10/2} = y^5\).
By utilizing this rule, we can simplify complex square roots containing variables, making them much easier to work with in algebraic expressions.
algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In our exercise, we use algebraic methods to simplify square roots involving both constants and variables.
Combining Simplified Components
Once we've simplified each part inside the square root \(\sqrt{169 w^{8} y^{10}}\), we combine them:
By breaking down the exercise into manageable steps, we use algebra to turn a complex expression into a simple, understandable one.
Combining Simplified Components
Once we've simplified each part inside the square root \(\sqrt{169 w^{8} y^{10}}\), we combine them:
- The square root of 169 is 13.
- The square root of \(w^{8}\) is \(w^4\).
- The square root of \(y^{10}\) is \(y^5\).
By breaking down the exercise into manageable steps, we use algebra to turn a complex expression into a simple, understandable one.