Add. Write your answer in simplest form.
$8 \sqrt{63}+10 \sqrt{7}$
Answer (2)
Simplify the radical 63 to 3 7 .
Substitute the simplified radical back into the first term: 8 63 = 24 7 .
Add the simplified terms: 24 7 + 10 7 .
Combine like terms: The final answer is 34 7 .
Explanation
Understanding the problem We are asked to add two terms involving square roots: 8 63 and 10 7 . To do this, we need to simplify the radicals and combine like terms.
Simplifying the radical First, we simplify the radical 63 . We look for perfect square factors of 63. Since 63 = 9 × 7 , we can rewrite 63 as 9 × 7 .
Rewriting the radical Now, we use the property a × b = a × b to get 9 × 7 = 9 × 7 = 3 7 .
Substituting back Substitute the simplified radical back into the first term: 8 63 = 8 ( 3 7 ) = 24 7 .
Adding the terms Now we can add the simplified first term to the second term: 24 7 + 10 7 .
Combining like terms Combine like terms by adding the coefficients: ( 24 + 10 ) 7 = 34 7 .
Final answer Therefore, the final answer is 34 7 .
Examples
Square roots often appear when calculating distances using the Pythagorean theorem. For example, if you're building a rectangular garden and want to find the length of the diagonal, you might end up needing to simplify and add square roots. Understanding how to manipulate these expressions allows you to accurately determine the required measurements, ensuring your garden fits perfectly in the space you have.
To add 8 63 + 10 7 , we first simplify 63 to 3 7 , making the expression 24 7 + 10 7 . Combining these gives us 34 7 . Thus, the final answer is 34 7 .
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