In the page "Exponent properties involving products" we are going to learn about how to multiply two or more terms involving exponents.

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

for example 5**³ = 5 x 5 x 5**

In words 5³ could be called as 5 to the power 3 or 5 cube**.**

**Rule 1:**

When we have to simplify two or more the terms which are multiplying with same base,then we have to put the same base and add the powers.

**Rule 2:**

Whenever we have two terms which are diving with the same base,we have to put only one base and we have to subtract the powers.

**Rule 3:**

Whenever we have power to the power,we have to multiply both powers.

**Rule 4:**

Anything to the power zero is 1.

**Rule 5:**

If we have same power for 2 or more terms which are multiplying or dividing,we have to apply the powers for every terms.

Note:

This rule is not applicable when two are more terms which are adding and subtracting.

For example (x + y) ^m = (x^m + y^m) is not correct

If the power goes from one side of equal sign to the other side,it will flip. that is x = 4² |

The other names of exponent are index and power.

Other things:

Point 1:

If we don't have any number in the power then we have to consider that there is 1

Point 2:

Incase we have negative power for any fraction and we want to make it as positive, we can write the power as positive and we should write its reciprocal only. For example

Find the value of (-5) **³**

To find value of the above expression, first we have to consider the power.

- If the power is odd, then the answer will have negative sign.
- If the power is even, then the answer will have positive sign.

here, the power of -5 is 3, that is odd. So, the answer will have negative sign and we have to multiply the base three times.

Finally we will get -125 as the answer.

**Procedure of multiplying two polynomials with exponents**

We will multiply two or more polynomials in the following order.

(1) Symbol

(2) Number

(3) Variable

Let us see how it works

Multiply ( 5 x² ) and (-2 x³)

= ( 5 x² ) **x** (-2 x³)

= 10 x⁵

**Question 1 :**

Simplify (5x²)⁴ x (2x)³

**Solution :**

= (5x²)⁴ x (2x)³

5⁴ = 5 x 5 x 5 x 5

2³ = 2 x 2 x 2

Since x⁸ and x³ are having same base, we can combine these terms.

Let us see the next example on "Exponent properties involving products"

**Question 2 :**

If 5 √5 x 5³= 5 ⁽ⁿ⁺²⁾ then find the value of n.

**Solution :**

**Let us see the next example on "Exponent properties involving products"**

**Question 3:**

Simplify (52 x⁶/13x⁻⁷)

**Solution :**

= (52 x⁶/13x⁻⁷)

We have negative power for the x term which is in the denominator.To make it as positive we are going to write it in numerator.

= (52 x⁶x⁷/13)

= 4 x⁽⁶⁺⁷⁾

= 4 x¹³

Let us see the next example on "Exponent properties involving products"

**Question 4 :**

Simplify (-7x²) (x⁴)

**Solution :**

= (-7x²) (x⁴)

= -7 x⁽²⁺⁴⁾

= -7 x⁶

**Question 5 :**

Simplify (10 x y³ z²) (-2 x y⁵z)

**Solution :**

= (10 x y³ z²) (-2 x y⁵z)

= -20 x^(1+1) y^(3+5) z^(2+1)

= -20 x²y⁸z³

**Question 6 :**

Simplify (-5 xyz) (4 y⁵z) y³

**Solution :**

= (-5 xyz) (4 y⁵z) y³

= -20 x y⁵y³z z

= -20 x y^(5+3) z^(1+1)

= - 20 x y⁸z²

**Question 7 :**

Simplify (7 a b² c³) (¾ a b c⁴)

**Solution :**

= (7 a b² c³) (¾ a b c⁴)

= (21/4) a^(1+1)b^(2+1)c^(3+4)

= (21/4) a²b³c⁷

**Question 8 :**

Simplify (-4 a³) (-5 a⁴)

**Solution :**

= (-4 a³) (-5 a⁴)

= 20 a^(3+4)

= 20 a⁷

**Question 9 :**

Simplify (2 a³) (¼) a⁴

**Solution :**

= (2 a³) (¼) a⁴

= (2/4)a^(3+4)

= (1/2) a⁷

**Question 10 :**

Simplify (10 p³) (7/25) p⁷

**Solution :**

= (10 p³) (7/25) p⁷

= 10(7/25)p^(3+7)

= (1/2) a¹⁰

After having gone through the stuff given above, we hope that the students would have understood "Exponent properties involving products".

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