Question

Please help! maths exponents question

Answer 1


`\frac{x^{(2)/(3)} + y^{(1)/(3)}}{x}`

Explanation:

First we start by factorising x out of the bottom expression. For the second term, this is simple, but for the first term, we need to make sure that when dividing by x^1, we subtract 1 from the exponent (so 5/3 - 1 = 2/3)

We now have:

`\frac{x^{(4)/(3)} - y^{(2)/(3)}}{x(x^{(2)/(3)} - y^{(1)/(3)})}`

On the bottom we have powers of 1/3 and 2/3, and on the top we have 4/3 and 2/3 which is exactly twice the powers on the bottom, and since a^2b ≡ (a^b)², we can rewrite the top as follows:

`\frac{(x^{(2)/(3)})^(2) - (y^{(2)/(3)})^(2)}{x(x^{(2)/(3)} - y^{(1)/(3)})}`

Using the difference of two squares on the numerator this factorises to:

`\frac{(x^{(2)/(3)} + y^{(1)/(3)})(x^{(2)/(3)} - y^{(1)/(3)})}{x(x^{(2)/(3)} - y^{(1)/(3)})}`

To finish we can cancel a bracket on the top and bottom to get:

`\frac{x^{(2)/(3)} + y^{(1)/(3)}}{x}`

Answer 2

`\large\text{$\frac{\left(x^{(2)/(3)}+y^{(1)/(3)}\right)}{x}$}`

Explanation:

Given rational expression:

`\large\text{$\frac{x^{(4)/(3)}-y^{(2)/(3)}}{x^{(5)/(3)}-xy^{(1)/(3)}}$}`

To simplify the given expression, we can first simplify the numerator and denominator separately.

Numerator

Rewrite the exponents of the terms in the numerator as:

`\large\text{$x^{\left((2)/(3)\cdot 2\right)}-y^{\left((1)/(3)\cdot 2\right)}$}`

`\textsf{Apply the exponent rule:} \quad a^(b\cdot c)=(a^b)^c`

`\large\text{$\left(x^{(2)/(3)}\right)^2-\left(y^{(1)/(3)}\right)^2$}`

Apply the difference of two squares: a² - b² = (a + b)(a - b)

`\large\text{$\left(x^{(2)/(3)}+y^{(1)/(3)}\right)\left(x^{(2)/(3)}-y^{(1)/(3)}\right)$}`

Denominator

Rewrite the exponent of the first term:

`\large\text{$x^{\left((3)/(3)+(2)/(3)\right)}-xy^{(1)/(3)}$}`

`\large\text{$x^{\left(1+(2)/(3)\right)}-xy^{(1)/(3)}$}`

`\textsf{Apply the exponent rule:} \quad a^(b+c)=a^b \cdot a^c`

`\large\text{$xx^{(2)/(3)}-xy^{(1)/(3)}$}`

Factor out the common term x:

`\large\text{$x\left(x^{(2)/(3)}-y^{(1)/(3)}\right)$}`

Therefore, after simplifying the numerator and denominator, the expression can be rewritten as:

`\large\text{$\frac{x^{(4)/(3)}-y^{(2)/(3)}}{x^{(5)/(3)}-xy^{(1)/(3)}}=\frac{\left(x^{(2)/(3)}+y^{(1)/(3)}\right)\left(x^{(2)/(3)}-y^{(1)/(3)}\right)}{x\left(x^{(2)/(3)}-y^{(1)/(3)}\right)}$}`

Notice that the numerator and denominator have a common factor.

Cancel the common factor:

`\large\text{$=\frac{\left(x^{(2)/(3)}+y^{(1)/(3)}\right)}{x}$}`

So, the simplified expression is:

`\large\boxed{\boxed{\frac{\left(x^{(2)/(3)}+y^{(1)/(3)}\right)}{x}}}`