Topic: Calculating With IndicesUse A Calculator To Work Out Each Of The Following:a) ${ 2 4 2^4 2 4 }$b) ${ 3 − 2 3^{-2} 3 − 2 }$c) { 0.23 − 3 0.23^{-3} 0.2 3 − 3 $}$d) { \left(\frac{2}{3}\right)^{-2} } E) ($\left(1

Introduction

Indices, also known as exponents, are a fundamental concept in mathematics that allows us to represent large numbers in a more compact and manageable form. In this article, we will explore the basics of indices and how to calculate with them using a calculator. We will work through a series of examples to demonstrate the application of indices in different scenarios.

What are Indices?

Indices are a shorthand way of representing repeated multiplication of a number. For example, ${2^4}$ can be read as "2 to the power of 4" or "2 multiplied by itself 4 times". This can be written as:

$$2^4 = 2 \times 2 \times 2 \times 2 = 16$$

Calculating with Indices

a) ${2^4}$

To calculate ${2^4}$ using a calculator, we simply enter the expression into the calculator and press the "=" button.

• Using a calculator: Enter 2^4 into the calculator and press "=".
• Result: The calculator will display the result as 16.

b) ${3^{-2}}$

To calculate ${3^{-2}}$ using a calculator, we need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${3^{-2}}$ is equivalent to ${\frac{1}{3^2}}$.

• Using a calculator: Enter 1/(3^2) into the calculator and press "=".
• Result: The calculator will display the result as 0.111111111.

c) ${[0.23^{-3}]}$

To calculate ${0.23^{-3}}$ using a calculator, we need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${0.23^{-3}}$ is equivalent to ${\frac{1}{0.23^3}}$.

• Using a calculator: Enter 1/(0.23^3) into the calculator and press "=".
• Result: The calculator will display the result as `0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
  Calculating with Indices: A Comprehensive Guide =====================================================

Q&A: Calculating with Indices

Q: What is the difference between a positive and negative exponent?

A: A positive exponent represents the base raised to that power, while a negative exponent represents the reciprocal of the base raised to the positive exponent. For example, ${2^4}$ is equivalent to ${2 \times 2 \times 2 \times 2}$, while ${2^{-4}}$ is equivalent to ${\frac{1}{2^4}}$.

Q: How do I calculate ${3^{-2}}$ using a calculator?

A: To calculate ${3^{-2}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${3^{-2}}$ is equivalent to ${\frac{1}{3^2}}$. You can enter 1/(3^2) into the calculator and press "=" to get the result.

Q: What is the value of ${0.23^{-3}}$?

A: To calculate ${0.23^{-3}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${0.23^{-3}}$ is equivalent to ${\frac{1}{0.23^3}}$. You can enter 1/(0.23^3) into the calculator and press "=" to get the result.

Q: How do I calculate ${\left(\frac{2}{3}\right)^{-2}}$?

A: To calculate ${\left(\frac{2}{3}\right)^{-2}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(\frac{2}{3}\right)^{-2}}$ is equivalent to ${\frac{1}{\left(\frac{2}{3}\right)^2}}$. You can enter 1/((2/3)^2) into the calculator and press "=" to get the result.

Q: What is the value of ${\left(1.2\right)^{-4}}$?

A: To calculate ${\left(1.2\right)^{-4}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(1.2\right)^{-4}}$ is equivalent to ${\frac{1}{\left(1.2\right)^4}}$. You can enter 1/(1.2^4) into the calculator and press "=" to get the result.

Q: How do I calculate ${\left(\frac{3}{4}\right)^{-3}}$?

A: To calculate ${\left(\frac{3}{4}\right)^{-3}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(\frac{3}{4}\right)^{-3}}$ is equivalent to ${\frac{1}{\left(\frac{3}{4}\right)^3}}$. You can enter 1/((3/4)^3) into the calculator and press "=" to get the result.

Q: What is the value of ${\left(2.5\right)^{-2}}$?

A: To calculate ${\left(2.5\right)^{-2}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(2.5\right)^{-2}}$ is equivalent to ${\frac{1}{\left(2.5\right)^2}}$. You can enter 1/(2.5^2) into the calculator and press "=" to get the result.

Q: How do I calculate ${\left(\frac{1}{2}\right)^{-4}}$?

A: To calculate ${\left(\frac{1}{2}\right)^{-4}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(\frac{1}{2}\right)^{-4}}$ is equivalent to ${\frac{1}{\left(\frac{1}{2}\right)^4}}$. You can enter 1/((1/2)^4) into the calculator and press "=" to get the result.

Q: What is the value of ${\left(3.2\right)^{-3}}$?

A: To calculate ${\left(3.2\right)^{-3}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(3.2\right)^{-3}}$ is equivalent to ${\frac{1}{\left(3.2\right)^3}}$. You can enter 1/(3.2^3) into the calculator and press "=" to get the result.

Q: How do I calculate ${\left(\frac{4}{5}\right)^{-2}}$?

A: To calculate ${\left(\frac{4}{5}\right)^{-2}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(\frac{4}{5}\right)^{-2}}$ is equivalent to ${\frac{1}{\left(\frac{4}{5}\right)^2}}$. You can enter 1/((4/5)^2) into the calculator and press "=" to get the result.

Q: What is the value of ${\left(1.5\right)^{-4}}$?

A: To calculate ${\left(1.5\right)^{-4}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(1.5\right)^{-4}}$ is equivalent to ${\frac{1}{\left(1.5\right)^4}}$. You can enter 1/(1.5^4) into the calculator and press "=" to get the result.

Q: How do I calculate ${\left(\frac{2}{5}\right)^{-3}}$?

A: To calculate ${\left(\frac{2}{5}\right)^{-3}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(\frac{2}{5}\right)^{-3}}$ is equivalent to ${\frac{1}{\left(\frac{2}{5}\right)^3}}$. You can enter 1/((2/5)^3) into the calculator and press "=" to get the result.

Q: What is the value of ${\left(2.8\right)^{-2}}$?

A: To calculate ${\left(2.8\right)^{-2}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(2.8\right)^{-2}}$ is equivalent to ${\frac{1}{\left(2.8\right)^2}}$. You can enter 1/(2.8^2) into the calculator and press "=" to get the result.

Q: How do I calculate ${\left(\frac{3}{2}\right)^{-4}}$?

A: To calculate ${\left(\frac{3}{2}\right)^{-4}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(\frac{3}{2}\right)^{-4}}$ is equivalent to ${\frac{1}{\left(\frac{3}{2}\right)^4}}$. You can enter 1/((3/2)^4) into the calculator and press "=" to get the result.

Q: What is the value of ${\left(4.5\right)^{-3}}$?

A: To calculate ${\left(4.5\right)^{-3}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(4.5\right)^{-3}}$ is equivalent to ${\frac{1}{\left(4.5\right)^3}}$. You can enter 1/(4.5^3) into the calculator and press "=" to get the result.

Q: How do I calculate ${\left(\frac{1}{3}\right)^{-2}}$?

A: To calculate ${\left(\frac{1}{3}\right)^{-2}}$ using a calculator, you need to understand that a negative exponent represents the reciprocal of the base raised to the positive exponent. In this case, ${\left(\frac{1}{3}\right)^{-2}}$ is equivalent to ${\frac{1}{\left(\frac{1}{3}\right)^2}}$. You can enter 1/((1/3)^2) into the calculator and press "=" to get the result.

Q: What is the value of ${\left(3.5\right)^{-4}}$?

A: To calculate ${\left(3.5\right)^{-4}}$