Problem 86
Question
Simplify. $$ \sqrt{48 m^{7} n^{5}} $$
Step-by-Step Solution
Verified Answer
The simplified form is \( 4 m^3 n^2 \sqrt{3mn} \).
1Step 1 - Factor inside the radical
Factor the number and the variables inside the square root, so that the factors can be easily simplified. For 48, the prime factorization is: \[ 48 = 2^4 \times 3 \] Combine this with the exponents of the variables: \[ 48 m^{7} n^{5} = 2^4 \times 3 \times m^7 \times n^5 \]
2Step 2 - Apply square root to each factor
Take the square root of each factor separately: \[ \ \sqrt{2^4 \times 3 \times m^7 \times n^5} = \sqrt{2^4} \times \sqrt{3} \times \sqrt{m^7} \times \sqrt{n^5} \]
3Step 3 - Simplify each square root
Simplify each of the square roots: \[ \ \sqrt{2^4} = 2^2 = 4 \ \sqrt{3} = \sqrt{3} \ \sqrt{m^7} = m^{3.5} = m^3 \sqrt{m} \ \sqrt{n^5} = n^{2.5} = n^2 \sqrt{n} \] So, the simplified form becomes: \[ 4 \sqrt{3} \times m^3 \sqrt{m} \times n^2 \sqrt{n} \]
4Step 4 - Simplify the expression
Combine all the simplified parts together: \[ 4 \times m^3 \times n^2 \times \sqrt{3} \times \sqrt{m} \times \sqrt{n} \] Combine like terms inside the square root: \[ = 4 m^3 n^2 \sqrt{3mn} \]
Key Concepts
Prime FactorizationExponentsSquare Root PropertiesSimplification Techniques
Prime Factorization
Understanding prime factorization is the key to simplifying square roots. Prime factorization involves breaking down a number into its smallest prime factors. For example, the number 48 can be expressed as the product of prime numbers: \(48 = 2^4 \times 3\).
When simplifying square roots, prime factorization helps identify which components can be easily simplified. Always factorize the number inside the square root first.
When simplifying square roots, prime factorization helps identify which components can be easily simplified. Always factorize the number inside the square root first.
Exponents
Exponents are essential when dealing with square roots of variables. An exponent tells you how many times a number, called the base, is multiplied by itself. For example, \(m^7\) means \(m\) is multiplied by itself 7 times.
When you take the square root of a number with an exponent, you essentially divide the exponent by 2. For example, \( \sqrt{m^7} = m^{7/2} \), which simplifies to \( m^{3.5} \) or \( m^3 \sqrt{m} \). This principle helps in simplifying complex square roots involving variables.
When you take the square root of a number with an exponent, you essentially divide the exponent by 2. For example, \( \sqrt{m^7} = m^{7/2} \), which simplifies to \( m^{3.5} \) or \( m^3 \sqrt{m} \). This principle helps in simplifying complex square roots involving variables.
Square Root Properties
Square root properties are rules that make it easier to simplify square roots. One such property is that the square root of a product is equal to the product of the square roots. For instance, \( \sqrt{48 m^{7} n^{5}} = \sqrt{48} \times \sqrt{m^7} \times \sqrt{n^5} \).
Another key property is that the square root of a power expression \(a^{2n}\) is \( a^n \). For example, \( \sqrt{2^4} = 2^2 = 4 \). These properties simplify the calculations and make the problem more manageable.
Another key property is that the square root of a power expression \(a^{2n}\) is \( a^n \). For example, \( \sqrt{2^4} = 2^2 = 4 \). These properties simplify the calculations and make the problem more manageable.
Simplification Techniques
Various simplification techniques are used to make expressions more straightforward. One critical technique is grouping terms under the square root to simplify them collectively. For example, \( \sqrt{3mn} \) can be handled as \( \sqrt{3} \times \sqrt{m} \times \sqrt{n} \).
Another technique is to always gather like terms together. The original expression \( \sqrt{48 m^{7} n^{5}} \) was simplified step by step by isolating factors and then taking their square roots individually. The final simplified form is \( 4 m^3 n^2 \sqrt{3mn} \). Techniques like these help in breaking down complex problems into simpler ones.
Another technique is to always gather like terms together. The original expression \( \sqrt{48 m^{7} n^{5}} \) was simplified step by step by isolating factors and then taking their square roots individually. The final simplified form is \( 4 m^3 n^2 \sqrt{3mn} \). Techniques like these help in breaking down complex problems into simpler ones.