Question

Can you teach me how to do these?? plzaa

Answer 1

If we are simplifying, ⅜ ab³a⁴ is
⅜a^5b³. when multiplying like bases you add exponents

second is xy²/2. when dividing like bases you subtract exponents

Answer 2

In general, you make use of the rules of exponents. It can be helpful to understand where they come from.

An exponent signifies repeated multiplication.

$x\cdot x\cdot x=x^{3}\qquad\text{the exponent 3 means x is a factor 3 times}$

When you multiply, you add exponents.

$(x\cdot x\cdot x)\times (x\cdot x)=(x\cdot x\cdot x\cdot x\cdot x)\\\\x^{3}\times x^{2}=x^{(3+2)}=x^{5}$

Likewise, when you divide, you subtract exponents. You can also think of this as adding the opposite of exponents that are in the denominator.

$\dfrac{x\cdot x\cdot x}{x\cdot x}=\dfrac{x\cdot x}{x\cdot x}\times x=x\\\\\dfrac{x^{3}}{x^{2}}=x^{(3-2)}=x^{1}=x$

It should be no surprise then that if there are excess factors in the denominator, they can be expressed using a negative exponent.

$\dfrac{x\cdot x}{x\cdot x\cdot x}=\dfrac{1}{x}\\\\\dfrac{x^{2}}{x^{3}}=x^{(2-3)}=x^{-1}\qquad\text{using exponents}$

The idea of using multiplication to show repeated addition applies to exponents as well.

$(x\cdot x)\times (x\cdot x)\times (x\cdot x)=(x\cdot x\cdot x\cdot x\cdot x\cdot x)\\\\=x^{2}\cdot x^{2}\cdot x^{2}=x^{(2+2+2)}\\\\=\left(x^{2}\right)^{3}=x^{2\cdot 3}=x^{6}$

_____

With these ideas in mind ...

$\frac{2}{3}ab^{3}a^{4}=\frac{2}{3}a^{(1+4)}b^{3}=\frac{2}{3}a^5b^3$

$\dfrac{8xy^{8}}{16y^{6}}=\frac{8}{16}xy^{(8-6)}=\frac{1}{2}xy^{2}$