Question

Simplify. ( 2 questions )

Answer 1

Solution :

$1.) \: \bf{y {}^{3} \: . \: y {}^{6} } \\ \\ \sf{ \red\bigstar \: \underline{Product \: rule \: : } } \\ \\ \boxed{\sf{a {}^{m} \: \times \: a {}^{n} \: = \: a {}^{m + n} }} \\ \\ \implies \: \sf{y {}^{3 \: + \: 6} } \\ \\ \implies \: \orange{\sf{y {}^{9 } } }$

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$2.) \: \bf{ \frac{a {}^{3} \: . \: a {}^{9 } }{a {}^{4} } } \\ \\ \implies \: \sf{ \dfrac{a {}^{3 + 9} }{a {}^{4} } } \\ \\ \implies \: \sf{ \dfrac{a {}^{12} }{a {}^{4} } } \\ \\ \sf{ \red\bigstar \: \underline{Quotient \: rule \: : } } \\ \\ \boxed{\sf{a {}^{m} \: \div \: a {}^{n} \: = \: a {}^{m - n} }} \\ \\ \implies \: \sf{a {}^{12 \: - \: 4} } \\ \\ \implies \: \orange{\sf{a {}^{8} }}$

Answer 2

Answer:

7)  y⁹

9)  a⁸

Step-by-step explanation:

when we multiply two exponents (of the same base), we can add them

so, a² · a³ = a² ⁺ ³ = a⁵

let's test it out!

(a = 2 in our example)

2² · 2³ = 2⁵

4 · 8  = 32   (2⁵ = 32)

32 = 32

because this is true, we know that we can add exponents of the same base when multiplying them together

now, let's try out the question at hand:

y³ · y⁶

well, we know that 3 + 6 = 9, so y³ · y⁶ can be simplified to y⁹

--

now, we're going to be working with division. We know that division is the inverse of multiplication, so we logically know that instead of adding exponents, we subtract them when dividing

here,

a³ ÷ a² would equal a¹    (3 - 2 = 1)

let's put a = 2 again:

a³ ÷ a² = a³ ⁻ ² = a¹ = a

2³ ÷  2² =   2¹

8  ÷   4   =  2

2   =  2

once again, we have proven this shortcut to be true

now, to the problem at hand!

first, we can multiply a³ · a⁹

a³  ·  a⁹   =  a³ ⁺ ⁹  =  a¹²

so, let's rewrite the problem:

a¹² /  a⁴

because we're dividing with exponents, we can subtract 12 - 4

a¹² / a⁴

a¹² ÷  a⁴

a¹² ⁻ ⁴

a⁸

so, our simplified answer is a⁸

(note: they must be the same base, [the thing that the exponent is attached to] for this to apply)

hope this helps (and makes sense)! have a lovely day :)