Problem 21
Question
Simplify each of the following expressions without using a calculator. $$3 \sqrt{25}+9 \sqrt{49}$$
Step-by-Step Solution
Verified Answer
The simplified expression is 78.
1Step 1: Simplify the Square Roots
First, simplify each square root in the expression. The square root of 25 is 5, because \(5^2 = 25\). Similarly, the square root of 49 is 7, because \(7^2 = 49\). So, \(\sqrt{25} = 5\) and \(\sqrt{49} = 7\).
2Step 2: Replace Square Roots with Their Values
Substitute the simplified square roots back into the original expression. Replace \(\sqrt{25}\) with 5, and \(\sqrt{49}\) with 7. The expression becomes \(3 imes 5 + 9 imes 7\).
3Step 3: Perform the Multiplications
Calculate the results of the multiplications. For \(3 imes 5\), the result is \(15\) and for \(9 imes 7\), the result is \(63\).
4Step 4: Add the Results
Finally, add the results of the multiplications together. \(15 + 63 = 78\).
Key Concepts
Square RootsMultiplication in AlgebraOrder of Operations
Square Roots
A square root helps you find a number which, when multiplied by itself, gives the original number. It is represented by the symbol \(\sqrt{}\). In the expression you are working with, you'll face two square roots: \(\sqrt{25}\) and \(\sqrt{49}\).
- For \(\sqrt{25}\), you might wonder what number times itself equals 25. That number is 5, because \(5 \times 5 = 25\). Therefore, \(\sqrt{25} = 5\).
- Likewise, to determine \(\sqrt{49}\), you're looking for a number that when multiplied by itself, yields 49. This number is 7 because \(7 \times 7 = 49\). Hence, \(\sqrt{49} = 7\).
Having a solid grasp of square roots is essential for simplifying expressions like the one you just worked on, where only basic multiplication is required after finding these values.
- For \(\sqrt{25}\), you might wonder what number times itself equals 25. That number is 5, because \(5 \times 5 = 25\). Therefore, \(\sqrt{25} = 5\).
- Likewise, to determine \(\sqrt{49}\), you're looking for a number that when multiplied by itself, yields 49. This number is 7 because \(7 \times 7 = 49\). Hence, \(\sqrt{49} = 7\).
Having a solid grasp of square roots is essential for simplifying expressions like the one you just worked on, where only basic multiplication is required after finding these values.
Multiplication in Algebra
Understanding multiplication in algebra is crucial for simplifying expressions. When you see a number placed next to a square root or any variable, it implies multiplication. Let's consider this in the context of your original expression, \(3 \sqrt{25} + 9 \sqrt{49}\).
Step 1 involved determining the values of the square roots, where \(\sqrt{25} = 5\) and \(\sqrt{49} = 7\). Now, the expression turns into multiplication: \(3 \times 5\) and \(9 \times 7\).
- For \(3 \times 5\), simply multiply to get 15.
- Similarly, \(9 \times 7\) is calculated as 63.
Once you've performed these multiplications, the next step becomes straightforward. With these skills, algebraic expressions involving multiplication become much easier to handle.
Step 1 involved determining the values of the square roots, where \(\sqrt{25} = 5\) and \(\sqrt{49} = 7\). Now, the expression turns into multiplication: \(3 \times 5\) and \(9 \times 7\).
- For \(3 \times 5\), simply multiply to get 15.
- Similarly, \(9 \times 7\) is calculated as 63.
Once you've performed these multiplications, the next step becomes straightforward. With these skills, algebraic expressions involving multiplication become much easier to handle.
Order of Operations
The order of operations is a set of rules to determine which calculations to perform first in a given mathematical expression. Remembering the acronym PEMDAS helps: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In the exercise, we converted \(3 \sqrt{25} + 9 \sqrt{49}\) into \(3 \times 5 + 9 \times 7\). According to PEMDAS, multiplication comes before addition.
- First, tackle the multiplications: \(3 \times 5 = 15\) and \(9 \times 7 = 63\).
- After completing these multiplications, proceed to addition: \(15 + 63 = 78\).
Understanding the order of operations ensures all parts of an expression are simplified correctly, leading to the right result without confusion.
In the exercise, we converted \(3 \sqrt{25} + 9 \sqrt{49}\) into \(3 \times 5 + 9 \times 7\). According to PEMDAS, multiplication comes before addition.
- First, tackle the multiplications: \(3 \times 5 = 15\) and \(9 \times 7 = 63\).
- After completing these multiplications, proceed to addition: \(15 + 63 = 78\).
Understanding the order of operations ensures all parts of an expression are simplified correctly, leading to the right result without confusion.
Other exercises in this chapter
Problem 20
Give the place value of the 5 in each of the following numbers. $$25.92$$
View solution Problem 21
Perform each of the following divisions. $$4 . 5 \longdiv { 2 9 . 2 5 }$$
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Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$4 \sqrt{20 x^{3}}+3 \sqrt{45 x^{3}}$$
View solution Problem 21
Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{32 x^{2}
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