Expression with Negative and Fractional Indices: Indices is the exponent or power which is raised to a variable or a number. For instance, in \(3^{5}, 5\) is the index of \(3\). The singular form of indices is the index. The index can be negative or fractional. In a fractional exponent, the exponent is always a fraction. Here, the numerator of the exponent is the power to which the number is being taken, and the denominator is the root that should be taken. For instance, \(27^{\frac{1}{3}}\) means take \(27\) to the third power and take the cube root of the result. In this article, we will understand expression with negative and fractional indices, the rules or the laws of the indices along with formulas and solved examples.

Indices

A variable or number could have an index. Index of a constant or the variable is a value that is raised to the power of the constant or the variable. The indices are also known as exponents or powers. It describes the number of times a given number must be multiplied. For example, in \(a^{m} a\) is the base, and \(m\) is the index. An index can be a positive, negative number, and fractional number.

We can say,

\(a^{m}=a \times a \times a \times \ldots \times a(\mathrm{m} \,\text {times})\)

For example, \(5 \times 5 \times 5\) can be written as \(5^{3}\) and it is read as \(5\) raised to the power \(3\) or \(3^{\text {rd }}\) power of \(5\). ln \(5^{3}\), we call \(5\) as the base and \(3\) as the index.

Example of Indices

We have learnt that if \(m\) is a positive integer and a non-zero number \(b\) is multiplied \(m\) times, then we get \(m\) index of \(b\) which is denoted by \(b^{m}\). Here \(b\) is called the base, \(m\) is called the index and \(b^{m}\) is called \(m t h\) index of \(b\) or \(b\) raised to index or power \(m\).

For example,

Mass of earth is \(5,970,000,000,000,000,000,000,000 \mathrm{~kg}\). Large numbers more conveniently using exponents, as, \(5.97 \times 10^{24} \mathrm{~kg}\)

We read \(10^{24}\) as \(10\) raised to the power \(24\).

We know \(7^{4}=7 \times 7 \times 7 \times 7\)

And \(3^{n}=3 \times 3 \times 3 \times 3 \times \ldots \times 3 \times 3 \ldots(n \,\text {times})\)

Note: Index \(2\) and index \(3\) have special names assigned to them.

POWER \(2\) is called SQUARE.

POWER \(3\) is called CUBE.

Example: \(10000=10 \times 10 \times 10 \times 10=10^{4}\)

The short notation \(10^{4}\) stands for the product \(10 \times 10 \times 10 \times 10\). Here \(10\) is called the base, and \(4\) is called an exponent.

We have used numbers like \(10,100,1000\), etc., while writing numbers in an expanded form.

Example: \(58761=5 \times 10000+8 \times 1000+7 \times 100+6 \times 10+1\)

Above expansion can be written as \(5 \times 10^{4}+8 \times 10^{3}+7 \times 10^{2}+6 \times 10^{1}+1\)

In the above instance, we have observed the numbers whose base is \(10\). However, the base could be any other number also.

Example: \(16=2 \times 2 \times 2 \times 2=2^{4}\). Here, \(2\) is the base and \(4\) is the exponent.

Let us see some examples: \(2^{5} \times 2^{3}, 3^{2} \times 3^{4}, \frac{4^{7}}{4^{4}},\left(2^{3}\right)^{2}\)

To find out the values of the above examples, we have some laws or rules of indices. Let us know those laws in the next part.

Indices Rules

Rules or laws of indices represent how to solve various types of mathematical expressions that contain indices and add, subtract, multiply and divide indices.

Rule 1: Multiplication of indices with a common base

The law implies that if the indices with the same bases are multiplied, then indices are added together.

The usual form of this law is \(a^{m} \times a^{n}=a^{m+n}\).

Rule 2: Dividing indices with the same base

In the division of index numbers with the same base, we need to do subtraction of indices.

The general form of this law is \(a^{m} \div a^{n}=a^{m-n}\).

Rule 3: The law of the index of an index

This law implies that we need to multiply the powers in case an exponential number is raised to another power.

The general form of this law is \(\left(a^{m}\right)^{n}=a^{m \times n}\).

Rule 4: The law of multiplication of powers with different bases but the same index.

This law states that we need to multiply the different bases to raise the same index to the product of bases.

The usual form of this law is \(a^{m} \times b^{m}=(a \times b)^{m}\).

Rule 5: The law of the division of powers with different bases but the same index.

This law states that we need to divide the different bases and raise the same index to the quotient of bases.

The usual form of this law is \(a^{m} \div b^{m}=\frac{a^{m}}{b^{m}}=\left(\frac{a}{b}\right)^{m}\).

Rule 6: The law of negative index.

When an index is negative, we change it to positive by taking the reciprocal of an index number.

The usual form of this law is \(a^{-n}=\frac{1}{a^{n}}\).

Rule 7: Zero power rule

If we consider any random base and if it is raised to the power of zero, it gives the result as one.

The usual form of this law is \(a^{0}=1\).

Rule 8: The law of fractional index.

When an index is a fraction, the numerator of the index is the power to which the number is being taken, and the denominator is the root that should be taken.

The usual form of this law is \(a^{\frac{1}{m}}=\sqrt[m]{a}\).

Problems based on Finding the Negative, Fractional Index of a Non-Zero Rational Number

Use the following outcomes wherever is needed.

If \(x\) is any integer and \(m\) is a whole number, then:

\(x^{m}=x \times x \times x \times x \times \ldots m \text { times }\)

\(x^{-m}=\frac{1}{x^{m}}=\frac{1}{x \times x \times x \times x \times \ldots m \text { times }}\)

1. If \(a\) and \(b\) are any two integers and \(m\) is a whole number, then:

\(\left(\frac{a}{b}\right)^{m}=\frac{a^{m}}{b^{m}}=\frac{a \times a \times a \times \ldots \text { times }}{b \times b \times b \times \ldots \text { times }}\), where \(b \neq 0\)

\(\left(\frac{a}{b}\right)^{-m}=\frac{b^{m}}{a^{m}}=\frac{b \times b \times b \times \ldots \text { times }}{a \times a \times a \times \ldots \text { times }}\), where \(a \neq 0\)

2. If \(x\) is a non-zero number, then \(x^{0}=1\)

3. If \(m\) and \(n\) are any two integers, then \(\frac{x^{m}}{x^{n}}=x^{m-n}, x \neq 0\)

4. If \(m\) is a non zero number, then \(a^{\frac{1}{m}}=\sqrt[m]{a}, a \neq 0\)

Solved Examples – Expression with Negative and Fractional Indices

Q.1. Express \(\left(\frac{3}{8}\right)^{-2} \times\left(\frac{4}{5}\right)^{-3}\) as a rational number of the form \(\frac{p}{q}\)
Ans:
We have, \(\left(\frac{3}{8}\right)^{-2} \times\left(\frac{4}{5}\right)^{-3}=\frac{1}{\left(\frac{3}{8}\right)^{2}} \times \frac{1}{\left(\frac{4}{5}\right)^{3}} \quad\left[\because a^{-n}=\frac{1}{a^{n}}\right]\)
\(=\frac{1}{\frac{3^{2}}{8^{2}}} \times \frac{1}{\frac{4^{3}}{5^{3}}} \quad\left[\because\left(\frac{a}{b}\right)^{n}=\left(\frac{a^{n}}{b^{n}}\right)\right]\)
\(=\frac{1}{\frac{9}{64}} \times \frac{1}{\frac{64}{125}}\)
\(=\frac{64}{9} \times \frac{125}{64}=\frac{125}{9}\)
Hence, \(\left(\frac{3}{8}\right)^{-2} \times\left(\frac{4}{5}\right)^{-3}=\frac{125}{9}\)

Q.2. Simplify \(\left(5^{-1} \times 3^{-1}\right)^{-1} \div 6^{-1}\)
Ans:
Given, \(\left(5^{-1} \times 3^{-1}\right)^{-1} \div 6^{-1}\)
\(=\left(\frac{1}{5} \times \frac{1}{3}\right)^{-1} \div \frac{1}{6} \quad \because(a)^{-1}=\frac{1}{a}\)
\(=\left(\frac{1}{5} \times \frac{1}{3}\right)^{-1} \div \frac{1}{6}\)
\(=\left(\frac{1}{15}\right)^{-1} \div \frac{1}{6}\)
\(=\frac{1}{\frac{1}{15}} \div \frac{1}{6}=15 \div \frac{1}{6}\)
\(=15 \times \frac{6}{1}=90\)
Hence, the value of \(\left(5^{-1} \times 3^{-1}\right)^{-1} \div 6^{-1}\) is \(90\).

Q.3. By what number should \((-6)^{-1}\) be multiplied so that the product may be equal to \((-3)^{-1}\) ?
Ans: Let \((-6)^{-1}\) be multiplied by \(x\) to get \((-6)^{-1}\)
Then, \(x \times(-6)^{-1}=(-3)^{-1}\)
\(\Rightarrow x=(-3)^{-1} \div(-6)^{-1}\)
\(\Rightarrow x=\left(\frac{1}{-3}\right) \div\left(\frac{1}{-6}\right)\)
\(\Rightarrow x=\left(\frac{1}{-3}\right) \times\left(\frac{-6}{1}\right)\)
\(\Rightarrow x=2\)
Hence, the required number is \(2\).

Q.4. Simplify \((125)^{\frac{1}{3}}\)
Ans: \((125)^{\frac{1}{3}}=\left(5^{3}\right)^{\frac{1}{3}}\)
We know that, \(\left(a^{m}\right)^{n}=a^{m \times n}\)
\(\left(5^{3}\right)^{\frac{1}{3}}=5\)

Q.5. Express \(2^{\frac{1}{7}}\) in root form.
Ans: The usual form of this law is \(a^{\frac{1}{m}}=\sqrt[m]{a}\).
Therefore, the root form of \(2^{\frac{1}{7}}\) is \(\sqrt[7]{2}\).

Summary

In the above article, we have discussed the definition of indices, different laws of indices. Also, this article explained the meaning of negative indices and the power rules. It explained the singular form of indices is the index. The index can be negative or fractional. In a fractional exponent, the exponent is always a fraction. Here, the numerator of the exponent is the power to which the number is being taken, and the denominator is the root that should be taken. At last, it shows some examples of negative and fractional indices.

Frequently Asked Questions (FAQs) – Expression with Negative and Fractional Indices

Q.1. Where do we use negative indices?
Ans: A positive index tells us how many times we need to multiply a base number, and a negative index tells us how many times we need to divide a base number. Also, we can write the negative index as \(a^{-n}=\frac{1}{a^{n}}\).

Q.2. What happens when we rise to a fractional index?
Ans: In a fractional exponent, the exponent is always a fraction. Here, the numerator of the exponent is the power to which the number is being taken, and the denominator is the root that should be taken. It means, if \(m\) is a non zero number, then \(a^{\frac{1}{m}}=\sqrt[m]{a}, a \neq 0\).

Q.3. How to write numbers in scientific notation?
Ans: We can write very small numbers in standard form, also known as scientific notation, by using the following steps:
(i) Get the number and see whether the number is between 1 and 10 or less than 1.
(ii) If the number belongs to between 1 and 10, then write down it as the product of the number itself and \(10^{\circ}\).
(iii) If the number is less than one, then we move the decimal point towards the right to just one digit on the left side of the decimal point. After that, we write the given number as the product of the number so obtained and \(10^{-n}\), where \(n\) is the number of places by which the decimal point has been moved to the right. The number we get is the standard form of the given number.

Q.4. What does 10 to the power negative 2 mean?
Ans: \(10^{-2}=\frac{1}{10^{2}}=\frac{1}{100}=0.01\)

Q.5. How do we know if the exponent is negative in scientific notation?
Ans: If we have the smaller number in the decimal form, i.e., smaller than 1. Then, the power is negative.

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