Arrange The Expressions In Increasing Order Of Their Values.$\[ \begin{array}{l} 10^0 + 10^1 + 1^{10} \\ 10^0 + 10^1 \times 1^{10} \\ 10^0 \times 10^1 \times 1^{10} \\ 10^0 \times 10^1 - 1^{10} \\ \end{array} \\]Place The Expressions In The

Introduction

In this article, we will explore the concept of arranging expressions in increasing order of their values. We will examine four different expressions and determine their values to arrange them in order from smallest to largest. This will involve applying mathematical operations and understanding the properties of exponents.

Understanding the Expressions

The four expressions we will be working with are:

1. $10^0 + 10^1 + 1^{10}$
2. $10^0 + 10^1 \times 1^{10}$
3. $10^0 \times 10^1 \times 1^{10}$
4. $10^0 \times 10^1 - 1^{10}$

Calculating the Values of the Expressions

Expression 1: $10^0 + 10^1 + 1^{10}$

To calculate the value of this expression, we need to evaluate each term separately.

• $10^0$ is equal to 1, since any number raised to the power of 0 is 1.
• $10^1$ is equal to 10, since any number raised to the power of 1 is itself.
• $1^{10}$ is equal to 1, since any non-zero number raised to any power is itself.

Therefore, the value of this expression is $1 + 10 + 1 = 12$.

Expression 2: $10^0 + 10^1 \times 1^{10}$

To calculate the value of this expression, we need to follow the order of operations (PEMDAS).

• $10^0$ is equal to 1.
• $10^1$ is equal to 10.
• $1^{10}$ is equal to 1.
• We multiply $10^1$ and $1^{10}$, which gives us $10 \times 1 = 10$.
• We add $10^0$ and the result of the multiplication, which gives us $1 + 10 = 11$.

Therefore, the value of this expression is 11.

Expression 3: $10^0 \times 10^1 \times 1^{10}$

To calculate the value of this expression, we need to follow the order of operations (PEMDAS).

• $10^0$ is equal to 1.
• $10^1$ is equal to 10.
• $1^{10}$ is equal to 1.
• We multiply $10^0$ and $10^1$, which gives us $1 \times 10 = 10$.
• We multiply the result of the first multiplication by $1^{10}$, which gives us $10 \times 1 = 10$.

Therefore, the value of this expression is 10.

Expression 4: $10^0 \times 10^1 - 1^{10}$

To calculate the value of this expression, we need to follow the order of operations (PEMDAS).

• $10^0$ is equal to 1.
• $10^1$ is equal to 10.
• $1^{10}$ is equal to 1.
• We multiply $10^0$ and $10^1$, which gives us $1 \times 10 = 10$.
• We subtract $1^{10}$ from the result of the multiplication, which gives us $10 - 1 = 9$.

Therefore, the value of this expression is 9.

Arranging the Expressions in Increasing Order of Their Values

Now that we have calculated the values of each expression, we can arrange them in increasing order of their values.

1. Expression 4: $10^0 \times 10^1 - 1^{10}$ = 9
2. Expression 3: $10^0 \times 10^1 \times 1^{10}$ = 10
3. Expression 2: $10^0 + 10^1 \times 1^{10}$ = 11
4. Expression 1: $10^0 + 10^1 + 1^{10}$ = 12

Therefore, the expressions in increasing order of their values are:

• $10^0 \times 10^1 - 1^{10}$
• $10^0 \times 10^1 \times 1^{10}$
• $10^0 + 10^1 \times 1^{10}$
• $10^0 + 10^1 + 1^{10}$

Conclusion

Frequently Asked Questions

In this article, we will answer some frequently asked questions related to arranging expressions in increasing order of their values.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed when there are multiple operations in an expression. The acronym PEMDAS stands for:

• Parentheses: Evaluate expressions inside parentheses first.
• Exponents: Evaluate any exponential expressions next (e.g., 2^3).
• Multiplication and Division: Evaluate any multiplication and division operations from left to right.
• Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I evaluate an expression with multiple operations?

A: To evaluate an expression with multiple operations, follow the order of operations (PEMDAS). For example, consider the expression 3 + 4 × 2. To evaluate this expression, we would follow the order of operations as follows:

1. Multiply 4 and 2: 4 × 2 = 8
2. Add 3 and 8: 3 + 8 = 11

Therefore, the value of the expression 3 + 4 × 2 is 11.

Q: What is the difference between an exponent and a power?

A: An exponent and a power are often used interchangeably, but technically, an exponent is the number that is raised to a power. For example, in the expression 2^3, the 3 is the exponent and the 2 is the base. The power is the result of raising the base to the exponent, which in this case is 2^3 = 8.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the following rules:

• When multiplying two numbers with the same base, add the exponents. For example, 2^3 × 2^4 = 2^(3+4) = 2^7.
• When dividing two numbers with the same base, subtract the exponents. For example, 2^4 ÷ 2^2 = 2^(4-2) = 2^2.
• When raising a power to a power, multiply the exponents. For example, (2³⁾4 = 2^(3×4) = 2^12.

Q: What is the difference between an expression and an equation?

A: An expression is a mathematical statement that contains variables and constants, but does not contain an equal sign (=). For example, 2x + 3 is an expression. An equation, on the other hand, is a mathematical statement that contains an equal sign (=) and is used to solve for a variable. For example, 2x + 3 = 5 is an equation.

Q: How do I solve an equation with multiple operations?

A: To solve an equation with multiple operations, follow the order of operations (PEMDAS) and then isolate the variable. For example, consider the equation 2x + 3 = 5. To solve for x, we would follow the order of operations as follows:

1. Subtract 3 from both sides: 2x = 5 - 3
2. Simplify the right-hand side: 2x = 2
3. Divide both sides by 2: x = 2/2
4. Simplify the right-hand side: x = 1

Therefore, the value of x is 1.

Conclusion

In this article, we have answered some frequently asked questions related to arranging expressions in increasing order of their values. We have covered topics such as the order of operations (PEMDAS), evaluating expressions with multiple operations, simplifying expressions with exponents, and solving equations with multiple operations. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of these concepts.