Evaluate Each Expression.a. $2\left(2^5\right$\]b. $(-10)^4(-10)^3$c. $\frac{\left(3^2\right)^7}{3^{10}}$d. $\left(10^2 \cdot 10^3\right)^2$

In mathematics, exponents and powers are used to represent repeated multiplication of a number. Understanding how to evaluate expressions with exponents and powers is crucial in solving mathematical problems. In this article, we will evaluate four different expressions using the rules of exponents and powers.

Expression a: $2\left(2^5\right)$

To evaluate this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the exponent $2^5$.
2. Multiply the result by 2.

The exponent $2^5$ means 2 multiplied by itself 5 times:

$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$

Now, multiply the result by 2:

$2 \times 32 = 64$

Therefore, the value of expression a is 64.

Expression b: $(-10)^4(-10)^3$

To evaluate this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the exponent $(-10)^4$.
2. Evaluate the exponent $(-10)^3$.
3. Multiply the results.

The exponent $(-10)^4$ means -10 multiplied by itself 4 times:

$(-10)^4 = (-10) \times (-10) \times (-10) \times (-10) = 10000$

The exponent $(-10)^3$ means -10 multiplied by itself 3 times:

$(-10)^3 = (-10) \times (-10) \times (-10) = -1000$

Now, multiply the results:

$10000 \times -1000 = -10000000$

Therefore, the value of expression b is -10000000.

Expression c: $\frac{\left(3^2\right)^7}{3^{10}}$

To evaluate this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the exponent $\left(3^2\right)^7$.
2. Evaluate the exponent $3^{10}$.
3. Divide the results.

The exponent $\left(3^2\right)^7$ means $3^2$ multiplied by itself 7 times:

$\left(3^2\right)^7 = (3^2) \times (3^2) \times (3^2) \times (3^2) \times (3^2) \times (3^2) \times (3^2) = 3^{14}$

The exponent $3^{10}$ means 3 multiplied by itself 10 times:

$3^{10} = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 59049$

Now, divide the results:

$\frac{3^{14}}{3^{10}} = 3^{14-10} = 3^4 = 81$

Therefore, the value of expression c is 81.

Expression d: $\left(10^2 \cdot 10^3\right)^2$

To evaluate this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the expression inside the parentheses.
2. Raise the result to the power of 2.

The expression inside the parentheses is $10^2 \cdot 10^3$:

$10^2 \cdot 10^3 = 10^2 \times 10^3 = 10^{2+3} = 10^5$

Now, raise the result to the power of 2:

$\left(10^5\right)^2 = 10^{5 \times 2} = 10^{10}$

Therefore, the value of expression d is $10^{10}$.

In the previous article, we evaluated four different expressions using the rules of exponents and powers. In this article, we will answer some frequently asked questions about evaluating expressions with exponents and powers.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the rule for multiplying exponents with the same base?

A: When we multiply exponents with the same base, we add the exponents. For example:

$2^3 \times 2^4 = 2^{3+4} = 2^7$

Q: What is the rule for dividing exponents with the same base?

A: When we divide exponents with the same base, we subtract the exponents. For example:

$\frac{2^5}{2^3} = 2^{5-3} = 2^2$

Q: What is the rule for raising a power to a power?

A: When we raise a power to a power, we multiply the exponents. For example:

$\left(2^3\right)^4 = 2^{3 \times 4} = 2^{12}$

Q: What is the rule for evaluating negative exponents?

A: When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example:

$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Q: What is the rule for evaluating fractional exponents?

A: When we have a fractional exponent, we can rewrite it as a root and a power. For example:

$2^{\frac{1}{2}} = \sqrt{2}$

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

A: To evaluate an expression with multiple exponents, we need to follow the order of operations. We start by evaluating any exponential expressions, then we evaluate any multiplication and division operations, and finally we evaluate any addition and subtraction operations.

For example, let's evaluate the expression:

$2^3 \times 2^4 + 3^2$

First, we evaluate the exponential expressions:

$2^3 = 8$ $2^4 = 16$ $3^2 = 9$

Next, we evaluate the multiplication operation:

$8 \times 16 = 128$

Finally, we evaluate the addition operation:

$128 + 9 = 137$

Therefore, the value of the expression is 137.

Q: How do I evaluate an expression with a negative exponent and a positive exponent?

A: To evaluate an expression with a negative exponent and a positive exponent, we need to follow the order of operations. We start by evaluating any exponential expressions, then we evaluate any multiplication and division operations, and finally we evaluate any addition and subtraction operations.

For example, let's evaluate the expression:

$2^{-3} \times 2^4$

First, we evaluate the exponential expressions:

$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ $2^4 = 16$

Next, we evaluate the multiplication operation:

$\frac{1}{8} \times 16 = 2$

Therefore, the value of the expression is 2.

In conclusion, evaluating expressions with exponents and powers requires following the order of operations and applying the rules of exponents and powers. By understanding these concepts, we can solve mathematical problems with ease.