diff --git a/CHANGELOG.md b/CHANGELOG.md
index 3134799f03..ad118e0e68 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -15,12 +15,14 @@ See [CHANGELOG - 6to5](CHANGELOG-6to5.md) for the pre-4.0.0 version changelog.
 
 ## 6.0.14
 
+ * **Spec Compliancy**
+  * Update exponentiation operator precedence.
+  * Fix parser bug where arrow functions have a higher precedence than they should.
  * **Bug Fix**
   * Fix SystemJS module formatter exporting function parameters.
   * Ensure that invalid identifier JSX attribute keys are quoted when transforming to calls.
   * Fix ES3 property literal plugin.
   * Fix parameters after defaults in arrow functions refering to the wrong `arguments`.
-  * Fix parser bug where arrow functions have a higher precedence than they should.
 
 ## 6.0.13
 
diff --git a/packages/babylon/src/parser/expression.js b/packages/babylon/src/parser/expression.js
index 677e78a1ed..6d2869f4b7 100644
--- a/packages/babylon/src/parser/expression.js
+++ b/packages/babylon/src/parser/expression.js
@@ -173,10 +173,18 @@ pp.parseExprOp = function(left, leftStartPos, leftStartLoc, minPrec, noIn) {
       let node = this.startNodeAt(leftStartPos, leftStartLoc);
       node.left = left;
       node.operator = this.state.value;
+
+      if (node.operator === "**" && left.type === "UnaryExpression" && left.extra && !left.extra.parenthesizedArgument) {
+        this.raise(left.argument.start, "Illegal expression. Wrap left hand side or entire exponentiation in parentheses.");
+      }
+
       let op = this.state.type;
       this.next();
-      let startPos = this.state.start, startLoc = this.state.startLoc;
+
+      let startPos = this.state.start;
+      let startLoc = this.state.startLoc;
       node.right = this.parseExprOp(this.parseMaybeUnary(), startPos, startLoc, op.rightAssociative ? prec - 1 : prec, noIn);
+
       this.finishNode(node, (op === tt.logicalOR || op === tt.logicalAND) ? "LogicalExpression" : "BinaryExpression");
       return this.parseExprOp(node, leftStartPos, leftStartLoc, minPrec, noIn);
     }
@@ -188,17 +196,26 @@ pp.parseExprOp = function(left, leftStartPos, leftStartLoc, minPrec, noIn) {
 
 pp.parseMaybeUnary = function (refShorthandDefaultPos) {
   if (this.state.type.prefix) {
-    let node = this.startNode(), update = this.match(tt.incDec);
+    let node = this.startNode();
+    let update = this.match(tt.incDec);
     node.operator = this.state.value;
     node.prefix = true;
     this.next();
+
+    let argType = this.state.type;
+    this.addExtra(node, "parenthesizedArgument", argType === tt.parenL);
     node.argument = this.parseMaybeUnary();
-    if (refShorthandDefaultPos && refShorthandDefaultPos.start) this.unexpected(refShorthandDefaultPos.start);
+
+    if (refShorthandDefaultPos && refShorthandDefaultPos.start) {
+      this.unexpected(refShorthandDefaultPos.start);
+    }
+
     if (update) {
       this.checkLVal(node.argument);
     } else if (this.state.strict && node.operator === "delete" && node.argument.type === "Identifier") {
       this.raise(node.start, "Deleting local variable in strict mode");
     }
+
     return this.finishNode(node, update ? "UpdateExpression" : "UnaryExpression");
   }
 
diff --git a/packages/babylon/test/fixtures/experimental/exponentiation-operator/options.json b/packages/babylon/test/fixtures/experimental/exponentiation-operator/options.json
new file mode 100644
index 0000000000..d01676b6ba
--- /dev/null
+++ b/packages/babylon/test/fixtures/experimental/exponentiation-operator/options.json
@@ -0,0 +1,3 @@
+{
+  "plugins": ["exponentiationOperator"]
+}
diff --git a/packages/babylon/test/fixtures/experimental/exponentiation-operator/unparenthesized-left/actual.js b/packages/babylon/test/fixtures/experimental/exponentiation-operator/unparenthesized-left/actual.js
new file mode 100644
index 0000000000..065fc4dab2
--- /dev/null
+++ b/packages/babylon/test/fixtures/experimental/exponentiation-operator/unparenthesized-left/actual.js
@@ -0,0 +1 @@
+-5 ** 6;
diff --git a/packages/babylon/test/fixtures/experimental/exponentiation-operator/unparenthesized-left/options.json b/packages/babylon/test/fixtures/experimental/exponentiation-operator/unparenthesized-left/options.json
new file mode 100644
index 0000000000..2c624e8cec
--- /dev/null
+++ b/packages/babylon/test/fixtures/experimental/exponentiation-operator/unparenthesized-left/options.json
@@ -0,0 +1,3 @@
+{
+  "throws": "Illegal expression. Wrap left hand side or entire exponentiation in parentheses. (1:1)"
+}