Problem 96
Question
Simplify the following problems. $$ 1^{8}+0^{10}+3^{2}\left(4^{2}+2^{3}\right) $$
Step-by-Step Solution
Verified Answer
$$
Answer: The simplified value of the given expression is 217.
1Step 1: Evaluate the powers
Evaluate the power of each number in the expression:
$$
1^8 = 1
$$
$$
0^{10} = 0
$$
$$
3^{2} = 9
$$
$$
4^{2} = 16
$$
$$
2^{3} = 8
$$
So, the expression becomes:
$$
1 + 0 + 9(16 + 8)
$$
2Step 2: Evaluate the expression inside the brackets
According to the BODMAS rule, evaluate the expression inside the brackets:
$$
16 + 8 = 24
$$
Now, the expression becomes:
$$
1 + 0 + 9(24)
$$
3Step 3: Perform the multiplication operation
Now, multiply 9 by the expression inside the brackets:
$$
9 \times 24 = 216
$$
So, the expression becomes:
$$
1 + 0 + 216
$$
4Step 4: Perform the addition operation
Finally, add all the numbers in the expression:
$$
1 + 0 + 216 = 217
$$
The simplified expression is:
$$
217
$$
Key Concepts
Powers and ExponentsBODMAS/BIDMAS RuleOrder of Operations
Powers and Exponents
Understanding powers and exponents is crucial to simplifying algebraic expressions effectively. A power is represented as a base number raised to an exponent. The exponent indicates how many times the base number is multiplied by itself. For example, in the expression \( 3^2 \), 3 is the base and 2 is the exponent, meaning \( 3 \times 3 \).
- **Calculating Powers and Exponents:** - An exponent of 1 means the number remains the same (e.g., \( 5^1 = 5 \)). - An exponent of 0 (except when the base is zero) equals 1, like \( 7^0 = 1 \). - For the original exercise given, calculate each term separately to simplify: \( 1^8 = 1 \), \( 0^{10} = 0 \), \( 3^2 = 9 \), \( 4^2 = 16 \), and \( 2^3 = 8 \).
Mastering the concept of powers and exponents makes simplifying expressions much smoother. This ensures each component of an expression is approached logically and methodically.
- **Calculating Powers and Exponents:** - An exponent of 1 means the number remains the same (e.g., \( 5^1 = 5 \)). - An exponent of 0 (except when the base is zero) equals 1, like \( 7^0 = 1 \). - For the original exercise given, calculate each term separately to simplify: \( 1^8 = 1 \), \( 0^{10} = 0 \), \( 3^2 = 9 \), \( 4^2 = 16 \), and \( 2^3 = 8 \).
Mastering the concept of powers and exponents makes simplifying expressions much smoother. This ensures each component of an expression is approached logically and methodically.
BODMAS/BIDMAS Rule
BODMAS/BIDMAS is a mnemonic that helps remember the order of operations used in mathematics. It stands for Brackets, Orders (i.e., powers and roots), Division, Multiplication, Addition, and Subtraction.
- **Application in Simplification:** - Start with any operations in **Brackets** first. In the given exercise, brackets signify the expression \( (4^2 + 2^3) \). - Then proceed to **Orders** or exponents, as previously calculated.- Always perform **Division** and **Multiplication** before moving on to **Addition** and **Subtraction**.
In our specific exercise, this rule directs us to first tackle what's inside the brackets, \( 16 + 8 \), resulting in 24. Next, apply multiplication with 9 before summing the final terms.
- **Application in Simplification:** - Start with any operations in **Brackets** first. In the given exercise, brackets signify the expression \( (4^2 + 2^3) \). - Then proceed to **Orders** or exponents, as previously calculated.- Always perform **Division** and **Multiplication** before moving on to **Addition** and **Subtraction**.
In our specific exercise, this rule directs us to first tackle what's inside the brackets, \( 16 + 8 \), resulting in 24. Next, apply multiplication with 9 before summing the final terms.
Order of Operations
The order of operations is a set of rules that determine the correct sequence in which to solve the different parts of a mathematical expression. This ensures consistency in solving expressions by prioritizing different operations.
- **Why Order Matters:** - Misplacing the order can alter results dramatically. For example, solving the expression from left to right without following order rules might yield incorrect results. - **Following the Correct Sequence:** - First, evaluate expressions within brackets. - Second, solve any power and roots as they follow directly after brackets. - Third, handle multiplication or division from left to right. - Lastly, manage any addition or subtraction.
In the step-by-step solution to the exercise, this approach was evident. Starting from focusing on exponents and proceeding logically through brackets, multiplication, and finally, addition, achieving the correct result of 217.
- **Why Order Matters:** - Misplacing the order can alter results dramatically. For example, solving the expression from left to right without following order rules might yield incorrect results. - **Following the Correct Sequence:** - First, evaluate expressions within brackets. - Second, solve any power and roots as they follow directly after brackets. - Third, handle multiplication or division from left to right. - Lastly, manage any addition or subtraction.
In the step-by-step solution to the exercise, this approach was evident. Starting from focusing on exponents and proceeding logically through brackets, multiplication, and finally, addition, achieving the correct result of 217.
Other exercises in this chapter
Problem 94
Simplify the following problems. $$ 4^{2}+8 $$
View solution Problem 95
Simplify the following problems. $$ 6^{3}+5(30) $$
View solution Problem 97
Simplify the following problems. $$ 12^{2}+0.3(11)^{2} $$
View solution Problem 98
Simplify the following problems. $$ \frac{3^{4}+1}{2^{2}+4^{2}+3^{2}} $$
View solution