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 self-titled. 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.