Using a Calculator to Evaluate Expressions

Oct 13, 2025 by pynkocean.com 141 views

What is the Base in the Expression $$7^4$$?

The base in the expression $$7^4$$ is 7. In mathematics, the base is the number that is being raised to a power or exponent. In this case, the base is 7 and the exponent is 4. The base is the foundation of the expression, and it determines the value of the expression when raised to the power of the exponent.

Evaluating Exponents

To evaluate an expression with an exponent, we need to multiply the base by itself as many times as the exponent indicates. For example, in the expression $$7^4$$, we need to multiply 7 by itself 4 times: $$7 \times 7 \times 7 \times 7 = 2401$$. This is the value of the expression $$7^4$$.

Using a Calculator to Evaluate Expressions

In addition to evaluating expressions by hand, we can also use a calculator to evaluate expressions with exponents. To use a calculator to evaluate an expression with an exponent, we simply enter the expression into the calculator and press the "calculate" or "evaluate" button. For example, to evaluate the expression $$5.2^3 \times 2.8^{-2}$$, we would enter the expression into the calculator and press the "calculate" button.

Evaluating Expressions with Negative Exponents

When evaluating expressions with negative exponents, we need to take the reciprocal of the base and change the sign of the exponent. For example, in the expression $$2.8^-2}$$, we need to take the reciprocal of 2.8 and change the sign of the exponent $$\frac{1{2.8^2} = \frac{1}{7.84} = 0.1276$$. We can then multiply this value by $$5.2^3$$ to get the final result.

Evaluating Expressions with Square Roots

In addition to evaluating expressions with exponents, we can also use a calculator to evaluate expressions with square roots. To evaluate an expression with a square root, we simply enter the expression into the calculator and press the "calculate" or "evaluate" button. For example, to evaluate the expression $$\sqrt{21.5} + 3.4^2$$, we would enter the expression into the calculator and press the "calculate" button.

Using a Calculator to Evaluate Expressions with Square Roots and Exponents

When evaluating expressions with both square roots and exponents, we need to follow the order of operations (PEMDAS). This means that we need to evaluate the expression inside the square root first, followed by the expression with the exponent. For example, in the expression $$\sqrt21.5} + 3.4^2$$, we need to evaluate the expression inside the square root first $$\sqrt{21.5 = 4.653$$. We can then evaluate the expression with the exponent: $$3.4^2 = 11.56$$. Finally, we can add these two values together to get the final result: $$4.653 + 11.56 = 16.213$$.

Conclusion

In conclusion, evaluating expressions with exponents and square roots requires a basic understanding of mathematical operations and the order of operations. By following the order of operations and using a calculator to evaluate expressions, we can quickly and accurately evaluate complex mathematical expressions.

Discussion Questions

What is the base in the expression $$8^5$$?
Use a calculator to evaluate $$4.5^2 \times 2.1^{-3}$$ (correct to 3 significant figures).
Use a calculator to evaluate $$\sqrt{36.2} + 2.5^3$$.

Answer Key

The base in the expression $$8^5$$ is 8.
The value of $$4.5^2 \times 2.1^{-3}$$ is 0.012.
The value of $$\sqrt{36.2} + 2.5^3$$ is 14.142.

Additional Resources

For more information on evaluating expressions with exponents and square roots, please see the following resources:

Khan Academy: Exponents and Exponential Functions
Mathway: Exponents and Exponential Functions
Wolfram Alpha: Exponents and Exponential Functions

References

"Algebra and Trigonometry" by Michael Sullivan
"Mathematics for the Nonmathematician" by Morris Kline
"Calculus" by Michael Spivak

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

A: An exponent is the number that is being raised to a power, while a power is the result of raising a number to an exponent. For example, in the expression $$7^4$$, 7 is the base (exponent) and 4 is the power.

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

A: To evaluate an expression with a negative exponent, you need to take the reciprocal of the base and change the sign of the exponent. For example, in the expression $$2.8^-2}$$, you would take the reciprocal of 2.8 and change the sign of the exponent $$\frac{1{2.8^2} = \frac{1}{7.84} = 0.1276$$.

Q: How do I evaluate an expression with a square root and an exponent?

A: To evaluate an expression with a square root and an exponent, you need to follow the order of operations (PEMDAS). This means that you need to evaluate the expression inside the square root first, followed by the expression with the exponent. For example, in the expression $$\sqrt21.5} + 3.4^2$$, you would evaluate the expression inside the square root first $$\sqrt{21.5 = 4.653$$, and then evaluate the expression with the exponent: $$3.4^2 = 11.56$$.

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

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when evaluating an expression. The acronym PEMDAS stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate expressions with exponents next.
Multiplication and Division: Evaluate multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate addition and subtraction operations from left to right.

Q: How do I use a calculator to evaluate an expression with exponents and square roots?

A: To use a calculator to evaluate an expression with exponents and square roots, you simply enter the expression into the calculator and press the "calculate" or "evaluate" button. For example, to evaluate the expression $$\sqrt{21.5} + 3.4^2$$, you would enter the expression into the calculator and press the "calculate" button.

Q: What is the difference between a base and a coefficient?

A: A base is the number that is being raised to a power, while a coefficient is a number that is multiplied by the base. For example, in the expression $$7 \times 4^2$$, 7 is the coefficient and 4 is the base.

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

A: To evaluate an expression with multiple exponents, you need to follow the order of operations (PEMDAS). This means that you need to evaluate the expressions with the highest exponents first, followed by the expressions with lower exponents. For example, in the expression $$2^3 \times 3^2$$, you would evaluate the expression with the highest exponent first: $$2^3 = 8$$, and then evaluate the expression with the lower exponent: $$3^2 = 9$$.

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

A: An exponent is a number that is being raised to a power, while a fraction is a number that is expressed as a ratio of two numbers. For example, in the expression $$\frac{1}{2^3}$$, 2 is the base (exponent) and 3 is the power, while 1 and 2 are the numerator and denominator of the fraction.

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

A: To evaluate an expression with a fraction and an exponent, you need to follow the order of operations (PEMDAS). This means that you need to evaluate the expression with the exponent first, followed by the fraction. For example, in the expression $$\frac1}{2^3}$$, you would evaluate the expression with the exponent first $$2^3 = 8$$, and then evaluate the fraction: $$\frac{1{8}$$.

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

A: An exponent is a number that is being raised to a power, while a logarithm is the inverse operation of an exponent. For example, in the expression $$\log_{10} 100$$, 10 is the base and 100 is the result of raising 10 to a power.

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

A: To evaluate an expression with a logarithm and an exponent, you need to follow the order of operations (PEMDAS). This means that you need to evaluate the expression with the logarithm first, followed by the expression with the exponent. For example, in the expression $$\log_10} 100 + 2^3$$, you would evaluate the expression with the logarithm first $$\log_{10 100 = 2$$, and then evaluate the expression with the exponent: $$2^3 = 8$$.