Today’s soundtrack is Dream Theater: Distance Over Time, a progressive metal album that I’m more excited about than I have been about anything in a long time. The album’s production is on point, the best it’s been since Mangini joined the crew. Myung’s bass sounds clear and beefy, and Mike’s snare sounds like a proper punch to the face, not like the downtuned mess on Dream Theater’s selftitled. Labrie’s voice is, as always, matchless. Petrucci and Rudess dance musically in intricate and creative ways. The album flows beautifully, never tripping up, always expanding and progressing 😉 . The lyrics are interesting; it’s one of the few albums that I’ve listened to in a while that has made me want to read the lyrics book along with the music. 10/10 from me.
Today, I’ll be reviewing the exponent laws. But first, let’s review powers. A power is comprised of a base and an exponent – for example, the power 3² has a base of three and an exponent of two. This simply means that we multiply 3 by 2 instances of itself – 3⋅3, which is 9.
Now, on to the laws of exponents. These laws only apply to powers with identical bases.

Product Law

When multiplying a power by another power, add the exponents

x³ ⋅ x² = x⁵


Quotient Law

When dividing a power by another power, subtract the exponents

x³ / x² = x¹


Power of a Power

When raising a power by an exponent, multiply the exponents

(x²)³ = x⁶


Power of a Product

When raising a product by an exponent, each term becomes a power

(xy)² = x²y²


Power of a Quotient

When raising a rational expression by an exponent, each term becomes a power

(x/y)² = x²/y²


Integral Exponent Rule

A power with a negative exponent has its exponent inverted and is then expressed as its own reciprocal

x⁻² = 1/x²

x² / x³ = x⁻¹ = 1/x


Rational Exponents

A base raised to a rational exponent becomes a radical with the index of the denominator; inside of the radical, the base is raised to value of the numerator

x^(2/3) = ³√(x²)

Whenever possible, we need to perform the math operations before applying the exponent laws.
If we are given a rational expression with inverse powers, we just switch their position from numerator to denominator or vice versa, inverting their sign as we go.
6a⁻² / b⁻³ = 6b³ / a⁻²
We may be asked to change a power to another base. What this means is that we must change the base value by modifying its exponents. For example, if we are asked to convert 27⁴ⁿ to base 3, we must find the exponent that we would apply to 3 to make it equal 27. If we cube 3, we get 27. So (3³)⁴ⁿ = 27. But we can go further! Using the Power of a Power law, we know to multiply our exponents together, giving us 3¹²ⁿ = 27⁴ⁿ as our final answer.