What is 2 + 2 and why

And another time-saving rule

If you want to multiply powers with different bases but the same exponent, you can first write them as products, rearrange the factors and then write the whole thing again as a power.

$$2^2*3^2 = 2 * 2*  3*3=2*3*2*3=(2*3)*(2*3)$$ $$=6*6=6^2 $$

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Swap order brackets

But it is also faster:

$$2^2*3^2=(2*3)^2=6^2$$

You can confirm the equality:
$$ 2 ^ 2 * 3 ^ 2 = 4 * 9 = 36 $$ and $$ 6 ^ 2 = 6 * 6 = 36 $$


Of course, this also works for variables:

$$ x ^ 3 * y ^ 3 = x * x * x * y * y * y = x * y * x * y * x * y $$

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Swap order

$$ = (x * y) * (x * y) * (x * y) $$ $$ = (x * y) ^ 3 $$
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cling

Or simply: $$ x ^ 3 * y ^ 3 = (x * y) ^ 3 $$

2nd power law - part 1
Do you want potencies with same exponent multiply, multiply the bases and keep the exponent unchanged.

$$ a ^ n * b ^ n = (a * b) ^ n $$

And with fractions

With the 2nd power law you also get a rule for dividing powers with the same exponent.

$$2^2:3^2 =2^2/3^2=(2*2)/(3*3)=2/3*2/3=(2/3)^2 $$

Or simply: $$ 2 ^ 2: 3 ^ 2 = 2 ^ 2/3 ^ 2 = (2/3) ^ 2 $$


You can confirm the equality:

$$ 2 ^ 2: 3 ^ 2 = 2 ^ 2/3 ^ 2 = 4/9 $$ and $$ (2/3) ^ 2 = 2/3 * 2/3 = 4/9 $$


It's the same for variables:

$$ x ^ 3: y ^ 3 = x ^ 3 / y ^ 3 = (x * x * x) / (y * y * y) = x / y * x / y * x / y = (x / y ) ^ 3 $$

Or simply: $$ x ^ 3: y ^ 3 = x ^ 3 / y ^ 3 = (x / y) ^ 3 $$

2nd power law - part 2
Do you want potencies with same exponent divide, divide the bases and keep the exponent unchanged.

$$ a ^ n: b ^ n = (a ^ n) / (b ^ n) = (a / b) ^ n = (a: b) ^ n $$

The following applies to the multiplication of fractions

$$ ("Numerator times numerator") / (\ text {Denominator times denominator $$

Working with tricks

Sometimes it is not very obvious how you use which rule in tasks. Then reformulate the term so that you can apply the rule well.

example 1:
$$2^2*3^(-2) =2^2*1/3^2=( 2*2)/(3*3)$$

$$= 2 * 2*  1/3*1/3=2*1/3*2*1/3=2/3*2/3=(2/3)^2 $$
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Reverse order rewrite

Or simply: $$ 2 ^ 2 * 3 ^ (- 2) = 2 ^ 2/3 ^ 2 = (2/3) ^ 2 $$

Write the task “appropriately” for the rule.


Example 2: With variables

Pretty awkward:
$$ x ^ 3: y ^ (- 3) = x ^ 3 * 1 / y ^ 3 = (x * x * x) * 1 / (y * y * y) $$

$$ = (x * x * x) / (y * y * y) = x / y * x / y * x / y = (x / y) ^ 3 $$

Or simply: $$ x ^ 3 * y ^ (- 3) = x ^ 3 / y ^ 3 = (x / y) ^ 3 $$

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And another trick!

You know the task: “Simplify as much as possible.” But you often don't know how to start. You can almost always apply math rules backwards and forwards.

example 1:

$$2^3*6^(-3) = 2^3/6^3=(2^3)/((2*3)^3)=(2^3)/(2^3*3^3)=1/3^3=1/27$$

To be able to simplify the term, you split $$ 6 = 2 * 3 $$ into factors. Then you can apply the 2nd power law backwards and then shorten it.

Example 2:

$$(2/3)^3*2^(-3)=2^3/3^3*1/2^3=2^3/(3^3*2^3)=1/3^3=1/27$$

Here you can use the 2nd power law for the division for the first factor $$ (2/3) ^ 3 $$ and the definition of powers with negative exponents for $$ 2 ^ (- 3) $$. Then you stick to the rules of fractions.

You can reduce a fraction by dividing the numerator and denominator by the same number.

Often times, when trying to simplify a term, it means shortening a fraction.

Cleverly combined!

If you are to simplify a term with powers, you need to know whether you can apply the first or the second power law. Or even both!

Hidden!

$$2^4/6^2 =2^4/(2*3)^2=2^4/(2^2*3^2)=2^4/2^2*1/3^2=2^(4-2)*1/3^2=2^2*1/3^2=4/9 $$

At first glance, neither of the two laws fits here. You use the fact that $$ 6 = 2 * 3 $$ is a product so that you can apply the 2nd power law - backwards - for the denominator of the fraction: $$ 6 ^ 2 = (2 * 3) ^ 2 = 2 ^ 2 * 3 ^ 2 $$.
If you've done that correctly, you can use the 1st power law to reduce it with $$ 2 ^ 2 $$. Then you just finish the math.