EXPONENTS WITH INTEGER BASES

Exponents with integer bases :

An integer is a number that can be written without a fractional component.

Before going to see example problems in this topic, first we have to see what is exponent and what is base. In this topic we will have four kind of questions,

• Positive base with positive exponent
• Positive base with negative exponent
• Negative base with positive exponent
• Negative base with negative exponent Positive base with positive exponent

The exponents of a number says, how many times to use the number in a multiplication.

For example, if we have 52 we have to multiply the base that is 5 two times.

52  =  5 x 5  =  25

Let us see some example problems based on the above concept.

Exponents with integer bases - Examples with Step by Step Solution

Example 1 :

Evaluate 10

Solution :

Base  =  10

Exponent  =  7

To evaluate this we have to multiply the base(10) seven times.

That is,

107 = 10 x 10 x 10 x 10 x 10 x 10 x 10

=  10000000

Hence the value of 10⁷ is 10000000.

Example 2 :

Evaluate 8²

Solution :

Base  =  8

Exponent  =  2

To evaluate this we have to multiply the base(8) two times.

That is,

8² = 8 x 8

=  64

Hence the value of 8² is 64.

Positive base with negative exponent

Whenever we have negative exponent for the positive base, first we have to make the power as positive, for that we have to write the reciprocal of base and change the negative power as positive.

For example,

5² =  (1/5)²  =  (1/5) x (1/5)  =  1/25

Let us see some example problem based on the above concept.

Example 3 :

Evaluate 4

Solution :

Base  =  4

Exponent  =  -4

To evaluate this first we have to make the power as positive

That is,

4 = (1/4)

Multiply 1/4 four times

=  (1/4) x (1/4) x (1/4) x (1/4)  =  1/256

Hence the value of 4 is 1/256.

Example 4 :

Evaluate 3²

Solution :

Base  =  3

Exponent  =  -2

To evaluate this first we have to make the power as positive

That is,

3² = (1/3)²

Multiply 1/3 two times

=  (1/3) x (1/3)  =  1/9

Hence the value of 3² is 1/9.

Negative base with positive exponent

The exponents of a number says how many times to use the number in a multiplication.

For example, if we have (-5)² we have to multiply the base that is -5 two times.

(-5)² = (-5) x (-5) = 25

Instead of repeating negative sign, we can follow a simple method to decide whether the answer will have positive sign or negative sign.

• If we have odd number as power then the answer will have negative sign.
• If we have even number as power then the answer will have positive sign.

Let us see some example problem based on the above concept.

Example 5 :

Evaluate (-7)

Solution :

Base  =  -7

Exponent  =  4

To evaluate this we have to multiply the base(-7) four times.

That is,

(-7) = (-7) x (-7) x (-7) x (-7)

=  2401

Since we have even power, the answer will have positive sign. It is enough to multiply 7 four times. Don't have to repeat negative sign.

Hence the value of (-7) is 2401

Example 6 :

Evaluate (-1)²

Solution :

Base  =  -1

Exponent  =  2

To evaluate this we have to multiply the base(-1) two times.

That is,

(-1)² = (-1) x (-1)

=  1

Since we have even power, the answer will have positive sign. It is enough to multiply 1 two times. Don't have to repeat negative sign.

Hence the value of (-1)² is 1.

Negative base with negative exponent

Whenever we have negative base and negative exponent, first we have to convert the negative exponent as positive. For that we have to take the reciprocal of base.

So, the power the will become positive. Now we have to check whether the power is odd or even.

• If it is odd the answer will have negative sign.
• If it is even the answer will have positive sign.

Let us see some example problem based on the above concept.

Example 7 :

Evaluate (-7)

Solution :

Base  =  -7

Exponent  =  -4

(-7)⁴  =  (-1/7)

Since the power is even the answer will have positive sign. Now we have to multiply 1/7 four times.

=  (1/7) x (1/7) x (1/7) x (1/7)  =  1/2401

Hence the value of (-7) is 1/2401.

Example 8 :

Evaluate (-6)³

Solution :

Base  =  -6

Exponent  =  -3

(-6)³  =  (-1/6)³

Since the power is odd, the answer will have negative sign. Now we have to multiply 1/6 two times.

=  (1/6) x (1/6) x (1/6)  =  -1/216

Hence the value of (-1/6)³ is -1/216. After having gone through the stuff given above, we hope that the students would have understood how to solve problems in exponents with integer bases.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

You can also visit our following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6 