Problem 90
Question
Find the value of \(\frac{4^{2}+(3+2)^{2}-1}{2^{3} \cdot 5}+\frac{2^{4}\left(3^{2}-2^{3}\right)}{4^{2}}\).
Step-by-Step Solution
Verified Answer
Answer: The value of the given expression is 2.
1Step 1: Simplify the expression within parentheses
According to the order of operations, simplify within the parentheses and brackets first. So, let's simplify \((3+2)^{2}\).
\((3+2)^{2} = 5^{2} = 25\)
Now the expression becomes: \(\frac{4^{2}+25-1}{2^{3}\cdot 5}+\frac{2^{4}\left(3^{2}-2^{3}\right)}{4^{2}}\)
2Step 2: Simplify the exponents and perform other arithmetic operations
Simplify any remaining exponents and perform other arithmetic operations in each part of the expression:
\(\frac{4^{2}+25-1}{2^{3}\cdot 5}+\frac{2^{4}\left(3^{2}-2^{3}\right)}{4^{2}}\)
\(\frac{16+25-1}{8\cdot 5}+\frac{16\left(9-8\right)}{16}\)
\(\frac{40}{40}+\frac{16}{16}\)
3Step 3: Perform the division and simplify the expression
Divide and simplify the expression:
\(\frac{40}{40}+\frac{16}{16}\)
\(1+1\)
4Step 4: Add the values
Add the values to get the final answer:
\(1+1=2\)
Therefore, the value of the given expression is 2.
Key Concepts
Order of OperationsSimplifying ExponentsArithmetic OperationsDivision in Algebra
Order of Operations
In algebra, when faced with a complex expression, it's crucial to follow the correct sequence of steps to simplify the expression. This sequence is known as the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
For example, given the expression \(\frac{4^{2}+(3+2)^{2}-1}{2^{3} \cdot 5}+\frac{2^{4}\left(3^{2}-2^{3}\right)}{4^{2}}\), the first step is to address any calculations inside parentheses - in this case, simplifying \(3+2\) to 5, then squaring it to 25. Following the correct order of operations ensures accuracy in problem solving and is the foundation of all algebraic simplification.
For example, given the expression \(\frac{4^{2}+(3+2)^{2}-1}{2^{3} \cdot 5}+\frac{2^{4}\left(3^{2}-2^{3}\right)}{4^{2}}\), the first step is to address any calculations inside parentheses - in this case, simplifying \(3+2\) to 5, then squaring it to 25. Following the correct order of operations ensures accuracy in problem solving and is the foundation of all algebraic simplification.
Simplifying Exponents
Understanding how to handle exponents—numbers that indicate how many times a number is multiplied by itself—is key in algebra. To simplify expressions with exponents, you apply the power first before any multiplication or division.
In our exercise, \(4^{2}\) is simplified to 16 and \(2^{3}\) to 8. Knowing exponent rules, such as \(a^{m} \cdot a^{n} = a^{m+n}\) and \(\frac{a^{m}}{a^{n}} = a^{m-n}\), is essential, as they allow for quick and accurate simplification of terms involving powers. This ensures calculations are made correctly within algebraic expressions.
In our exercise, \(4^{2}\) is simplified to 16 and \(2^{3}\) to 8. Knowing exponent rules, such as \(a^{m} \cdot a^{n} = a^{m+n}\) and \(\frac{a^{m}}{a^{n}} = a^{m-n}\), is essential, as they allow for quick and accurate simplification of terms involving powers. This ensures calculations are made correctly within algebraic expressions.
Arithmetic Operations
Once exponents are simplified, arithmetic operations—addition, subtraction, multiplication, and division—come into play. In handling the example provided, after the exponents are simplified, you proceed with multiplying \(16\) by \(9-8\) and dividing the results by \(16\), and also dividing \(40\) by \(40\). These steps follow a systematic process which is crucial in solving algebra problems.
Arithmetic in algebra is similar to basic arithmetic, yet it often involves unknowns (variables) and can be subject to additional rules, such as those governing the distribution of multiplication over addition. Correct application of these operations is vital for reaching an accurate solution.
Arithmetic in algebra is similar to basic arithmetic, yet it often involves unknowns (variables) and can be subject to additional rules, such as those governing the distribution of multiplication over addition. Correct application of these operations is vital for reaching an accurate solution.
Division in Algebra
Division plays a distinct role in algebra, particularly in simplifying expressions. It can be seen as the inverse of multiplication and comes with its own set of rules. For instance, any number divided by itself equals 1, as shown in our problem with \(\frac{40}{40}\) and \(\frac{16}{16}\).
Understanding how to divide terms, especially when variables are involved, is fundamental. In many cases, simplification involves reducing expressions to their simplest form by canceling out like terms. It's important to be comfortable with division, as it is a frequent pitfall for students. Mastery of division in algebra is a cornerstone for progressing in mathematics.
Understanding how to divide terms, especially when variables are involved, is fundamental. In many cases, simplification involves reducing expressions to their simplest form by canceling out like terms. It's important to be comfortable with division, as it is a frequent pitfall for students. Mastery of division in algebra is a cornerstone for progressing in mathematics.
Other exercises in this chapter
Problem 89
Find the value of \(4^{2}+3^{2} \cdot 2^{3}-10 \cdot 8\).
View solution Problem 90
For the following problems, expand the terms so that no exponents appear. $$ \left(6 a^{2}\right)^{3}(5 c-4)^{2} $$
View solution Problem 91
For the following problems, expand the terms so that no exponents appear. $$ \left(x^{3}+7\right)^{2}\left(y^{2}-3\right)^{3}(z+10) $$
View solution Problem 92
Choose values for \(a\) and \(b\) to show that a. \((a+b)^{2}\) is not always equal to \(a^{2}+b^{2}\). b. \((a+b)^{2}\) may be equal to \(a^{2}+b^{2}\).
View solution